Most all of the images in the book of Revelation are referring to similar images in the Old Testament, so it seems the author of the visions was trying to cause the intended audience to think about each of the old testament visions and what they were referring to. Places where it says things like "to him who has understanding..." seem to indicate that some of the audience would understand the references whereas others would not. Perhaps those who would understand this reference to "666" would be those who knew and understood its reference in the Old Testament.
The "666" quote in Revelation 13 & 14:
"15And it was given to him to give breath to the image of the beast, so that the image of the beast would even speak and cause as many as do not worship the image of the beast to be killed. 16And he causes all, the small and the great, and the rich and the poor, and the free men and the slaves, to be given a mark on their right hand or on their forehead, 17and he provides that no one will be able to buy or to sell, except the one who has the mark, either the name of the beast or the number of his name. 18Here is wisdom Let him who has understanding calculate the number of the beast, for the number is that of a man; and his number is six hundred and sixty-six."
Note that "666" is "the number of a man", so the question naturally arises: "Who is that man?". While the passage could have more than one intended reference, at very least the passage would evoke in the educated Jewish reader any significant instances of that number in the old testament. So lets take a look at those references.
Old Testament References to "666"
There are two references where "666" is mentioned as part of a list of people returning from Babylon:
Neh 7:17-19 : ...17the sons of Azgad, 2,322; 18the sons of Adonikam, 667; 19the sons of Bigvai, 2,067;...
Since there is no reference to anything historically significant done by the sons of Adonikam and it is unclear whether 666 or 667 people from this family returned from Babylon, this seems to be a chance instance of where there was a group of 666 or 667 people and that probably doesn't relate to the book of Revelation.
The only other reference to "666" in the Bible is in two passages describing the same events, one in Chronicles and one in Kings:2 Chr 9:13-14;9:22-28
13Now the weight of gold which came to Solomon in one year was 666 talents of gold (about 50,000 pounds of gold), 14besides that which the traders and merchants brought; and all the kings of Arabia and the governors of the country brought gold and silver to Solomon.... 22So King Solomon became greater than all the kings of the earth in riches and wisdom. 23And all the kings of the earth were seeking the presence of Solomon, to hear his wisdom which God had put in his heart. 24They brought every man his gift, articles of silver and gold, garments, weapons, spices, horses and mules, so much year by year. 25Now Solomon had 4,000 stalls for horses and chariots and 12,000 horsemen, and he stationed them in the chariot cities and with the king in Jerusalem. 26He was the ruler over all the kings from the Euphrates River even to the land of the Philistines, and as far as the border of Egypt. 27The king made silver as common as stones in Jerusalem, and he made cedars as plentiful as sycamore trees that are in the lowland. 28And they were bringing horses for Solomon from Egypt and from all countries.
So, Chronicles shows how Solomon gathered gold and horses, but doesn't mention anything about Solomon in a negative light (such as having multiple wives and honoring other gods). To see the full picture of what happened we have to look at the book of Kings:
14Now the weight of gold which came in to Solomon in one year was 666 talents of gold, 15besides that from the traders and the wares of the merchants and all the kings of the Arabs and the governors of the country.... 23So King Solomon became greater than all the kings of the earth in riches and in wisdom. 24All the earth was seeking the presence of Solomon, to hear his wisdom which God had put in his heart. 25They brought every man his gift, articles of silver and gold, garments, weapons, spices, horses, and mules, so much year by year. 26Now Solomon gathered chariots and horsemen; and he had 1,400 chariots and 12,000 horsemen, and he stationed them in the chariot cities and with the king in Jerusalem. 27The king made silver as common as stones in Jerusalem, and he made cedars as plentiful as sycamore trees that are in the lowland...
1Now King Solomon loved many foreign women along with the daughter of Pharaoh: Moabite, Ammonite, Edomite, Sidonian, and Hittite women, 2from the nations concerning which the LORD had said to the sons of Israel, "You shall not associate with them, nor shall they associate with you, for they will surely turn your heart away after their gods." Solomon held fast to these in love. 3He had seven hundred wives, princesses, and three hundred concubines, and his wives turned his heart away. 4For when Solomon was old, his wives turned his heart away after other gods; and his heart was not wholly devoted to the LORD his God, as the heart of David his father had been. 5For Solomon went after Ashtoreth the goddess of the Sidonians and after Milcom the detestable idol of the Ammonites. 6Solomon did what was evil in the sight of the LORD, and did not follow the LORD fully, as David his father had done. 7Then Solomon built a high place for Chemosh the detestable idol of Moab, on the mountain which is east of Jerusalem, and for Molech the detestable idol of the sons of Ammon. 8Thus also he did for all his foreign wives, who burned incense and sacrificed to their gods. 9Now the LORD was angry with Solomon because his heart was turned away from the LORD, the God of Israel, who had appeared to him twice, 10and had commanded him concerning this thing, that he should not go after other gods; but he did not observe what the LORD had commanded. 11So the LORD said to Solomon, "Because you have done this, and you have not kept My covenant and My statutes, which I have commanded you, I will surely tear the kingdom from you, and will give it to your servant. 12"Nevertheless I will not do it in your days for the sake of your father David, but I will tear it out of the hand of your son. 13"However, I will not tear away all the kingdom, but I will give one tribe to your son for the sake of My servant David and for the sake of Jerusalem which I have chosen."
So, it seems that Solomon increased in military might, inter-married with rulers of other countries, gathered up lots of wealth, and he honored other Gods from countries around him rather than exclusively honoring the creator. Interestingly, those are the key things a world ruler would need to do to gain power and maintain peace. But how does that compare to what the Pentateuch says a king of Israel should do?
What the Creator says kings should/should not do
14"When you enter the land which the LORD your God gives you, and you possess it and live in it, and you say, 'I will set a king over me like all the nations who are around me,' 15you shall surely set a king over you whom the LORD your God chooses, one from among your countrymen you shall set as king over yourselves; you may not put a foreigner over yourselves who is not your countryman. 16"Moreover, he shall not multiply horses for himself, nor shall he cause the people to return to Egypt to multiply horses, since the LORD has said to you, 'You shall never again return that way.' 17"He shall not multiply wives for himself, or else his heart will turn away; nor shall he greatly increase silver and gold for himself. 18"Now it shall come about when he sits on the throne of his kingdom, he shall write for himself a copy of this law on a scroll in the presence of the Levitical priests. 19"It shall be with him and he shall read it all the days of his life, that he may learn to fear the LORD his God, by carefully observing all the words of this law and these statutes, 20that his heart may not be lifted up above his countrymen and that he may not turn aside from the commandment, to the right or the left, so that he and his sons may continue long in his kingdom in the midst of Israel.
So, Solomon was the wisest and most powerful king of his time, but he disregarded what the Pentateuch said he should not do as king. Basically, he thought he knew a better way to rule than the one given by the designer and creator of the universe, and his methods were seemingly successful. So, that fits very well with as an illustration of a coming set of future world leaders, who believe that they need to use various methods of obtaining power in order to maintain peace.
Why don't students of the Bible see this seemingly obvious connection?
I think people don't focus on this connection because of the common view that "Evil" is the opposite of "Good". Many people would say Solomon is described as wise and good and so he couldn't possibly be the intended reference concerning a coming world leader who is considered by the Bible to be evil. But in fact normally the goal of someone who is "Evil" is actually trying to be better and smarter, not worse and more destructive, than what the creator designed. For example, Hitler thought he was helping the evolution of European races by getting rid of inferior individuals and races. People who kill unborn babies think they are helping young mothers be more successful in the formative years of their lives. Eve thought it would be best to eat the fruit rather than follow God's recommendation. And, most likely, Satan himself thinks he can do better than the creator's way. So, evil normally starts out thinking that
it is working to do better than the method that the creator designed, but very bad things are caused by evil people as the unintended result of their belief system (the "necessary evil" to get to their Utopian system).
Here is a test. Which one of these pictures pictures evil and which one pictures good? The answer is that we can't know by looking. Many times good can look like evil (such as tough law enforcement and capital punishment) and many times evil can look like good (such as government subsidies, which steal from one person to give to another, and interfaith prayer), "for even Satan disguises himself as an angel of light. (2 Cor 11:14)"
So, if this interpretation of the "666" reference is correct, then apparently it implies that a set of world leaders will arise that will seem to be as wise and respected as Solomon.They will work to bring peace and prosperity to the world. Of necessity, they will amass wealth, build a strong military, and promote inter-religious cooperation (and possibly intermarry with other powerful families). They will also, at some point, limit the ability to buy or sell as yet another means of necessary power and control. In the end they will end up believing that they need to kill those who do not honor the laws of their government above conflicting laws given by the creator.
So, don't be looking for a world leader who looks evil. But rather keep a look out for world leaders with true wisdom, but who value peace and safety more than they value either truth or the one true designer/creator (who is the only one who can actually guide us into truth and peace).