How did fallen angels impregnate women and make Nephilim? And, why did they do this?
In Genesis 6 we learn that the primary cause of the flood was the emergence of a new, insidious threat to God's plan of redemption. Fallen angels, called "sons of God" (ben Elohim) in Genesis 6, were taking women, called "daughters of men," for sexual relationships. The result of these unions were a grotesque being scripture calls Nephilim. The Nephilim were giants, people who were grossly distorted by their unnatural origins.
Gen. 6:1 Now it came about, when men began to multiply on the face of the land, and daughters were born to them,
Gen. 6:2 that the sons of God saw that the daughters of men were beautiful; and they took wives for themselves, whomever they chose.
Gen. 6:4 The Nephilim were on the earth in those days, and also afterward, when the sons of God came in to the daughters of men, and they bore children to them. Those were the mighty men who were of old, men of renown.
Satan instigated this plan hoping to polute the human "seed" to stop God's plan to bring forth a Messiah, the Seed, promised to come one day and destroy (i.e., bruise) Satan:
Gen. 3:15 And I will put enmity
Between you and the woman,
And between your seed and her seed;
He shall bruise you on the head,
And you shall bruise him on the heel.”
Clearly, the Lord could not stand by and allow such an attempt to succeed. Peter also refers to this occasion in his second letter in explaining the Lord's response:
2Pet. 2:4 For if God did not spare angels when they sinned, but cast them into hell and committed them to pits of darkness, reserved for judgment;
2Pet. 2:5 and did not spare the ancient world, but preserved Noah, a preacher of righteousness, with seven others, when He brought a flood upon the world of the ungodly;
The Lord punished the demons and destroyed the fruit of their evil labors. Clearly, the act of mating happened, and the demons involved in this sin paid a dear price, but apart from these references, we have little to help us understand how spiritual creatures can mate with physical beings. We do not know specifically how the demons impregnated women, but we do know of other circumstances when angels appear to men in the form of bodies. For example, Abraham is met by three "men" in Genesis 18:
Gen. 18:2 When he lifted up his eyes and looked, behold, three men were standing opposite him; and when he saw them, he ran from the tent door to meet them and bowed himself to the earth,
Later in the chapter, we observe these men eating with Abraham and speaking with him, yet they were not men. One was the Lord Himself and the other two were angels. So clearly, angels can take the form of men when desired.
Secondly, we know that the Holy Spirit impregnated Mary. While we're not suggesting that demons' abilities are equal to God's, nevertheless it demonstrates that Spirit and flesh can interact in ways we don't completely understand.
So although the details are not provided for us, scripture testifies that demons and women mated in some way to produce a race of creatures called Nephalim.
You may also enjoy the following article: How did the ten spies see Nephilim after the Flood?