What simply happened was that the serpent mated with Eve, and, possibly, other women of the day, and this produced children. The serpent has many names: Nachash, Azazel and Amalek, for example. What began from these unions formed individual groups, or nations, of people, such as the Biblical Amalekites.
Although, in the end, there is nothing wrong with groups of people mixing together, what was happening at the time of these fallen angels posed major problems. The Nephilim, upon helping to overthrow Adam's authority, were now involved in other ways of control and manipulation. The women of the day became seduced by their overwhelming stature, charm and knowledge, and were being sexually exploited, as well. Like Eve, children were born unto them, which are thought of, today, as the Biblical giants. The sons of Cain, or Cainites, also mingled with other people, and the Nephilim, which increased the sinful ways of the people. What began a fairly simple and orderly world was now becoming very complex. Approximately 500 years after the creation of Adam, the Nephilim began to have these hybrid humans:
"First, they were called Nefilim, the fallen, at the time they were dropped down from heaven. After they joined up with the females of human
kind and had children from them, the children were called Anakim."
The Zohar 40 Shlach Lecha 10
This is echoed in Genesis:
Genesis 6:1 "And it came to pass, when man began to multiply on the face of the earth, and daughters were born unto them,
6:4 There were giants (Nephilim) on the earth in those days ..."
Intermingled in these confusing verses of Genesis, we have the mention of another group of individuals bent on the same outcome as these Nephilim. The Nephilim fell because they were jealous of Adam, around the time of the Garden, and wanted to bring him down. Almost 500 years after the Nephilim first cohabitated with women, the Watchers also began to crumble the morality of the world. Lets see how.
The Nephilim were endowed with a great amount of previously unheard-of knowledge: herbalism, divination, astrology, etc. They used this knowledge to seduce women into fulfilling their earthly, sexual desires. The Watchers were another group of large angels (which are also mentioned in Daniel 4:13 and 17), apparently unscathed by this first fall of Adam. They were about to come to the same fate as the Nephilim.
The Watchers (or "sleepless" angels) were assigned to "watch" over mankind, and help them in positive ways. They had the tasks of overseeing and assisting human beings in a lot of their worldly decisions. They noticed, however, how sinful mankind had become, because of the departure of morality now present in the world. They, like the Nephilim, began to complain to God:
"And God said unto them... 'I created him out of the dust... and I will not make my handiwork a laughingstock for his enemies.' And those angels
said... 'knowest that we have reviled Adam.' And God answered them on behalf of Adam... 'You have I created out of fire and air with the one
intent (that ye should) praise (Me). Him have I created of twice as many elements as you - of dust and water, and of wind and fire... And if his
heart inciteth him to good, he walketh with good intent; and if the devil seduceth him, he walketh with him on an evil path. As for you, ye have
no other object in your minds but praise of Me... And now, why do ye magnify yourselves above Adam? If ye were as he is... ye would have
transgressed My commandment more than he hath done...' And the angels said unto Him... 'we will not oppose Thy word; for we are spiritual
beings for life, and he is a creature of dust (doomed) to folly...' And straitway there were given unto them... flesh, and blood, and a heart of
the children of men."
Kerba Nagast: 100 Concerning the Angels
Apparently, the Watchers wanted to prove something to God. They apparently thought, with their superior intellect, they would not have the ignorance to do anything like any of their predecessors. Even though they often descended to the earth, and taught people upright and worthy sentiments, they did not understand how temptation could change those under it. When God allowed them to descend, they would not fair much better.
Which brings us to the second part of Genesis Six, a confusing verse, without our understanding of the above. With all of this information, however, it makes sense:
Genesis 6:2 "That the sons of God (the Watchers) saw the daughters of men that they were fair, and took them wives of all they chose.
6:4 There were giants (the Nephilim) in the earth in those days (of the Watcher's descent); and also after that (after the time of their
descent), when the sons of God (the Watchers) came into the daughters of men, and they bare children to them, the same (children
of the Watchers) became the mighty men which were of old (or, from those angels of old), men of renown (or, they were men who
'made a name for themselves')."
Even the Watchers blew it! Instead of showing God they could handle the world, they fell, just like the Nephilim did. That is why they are lumped together with them in these verses. They also fell to the temptations of these women, and bore children to them. They also taught these women and others more of those previously forbidden pieces of knowledge, in exchange for sexual favors. Immorality was everywhere, thanks to their sacrilegious and indecent behaviors.