There's an incredible amount of information that those "begats" give the reader. There are 42 generations, broken into a trinity of 14s. The numerological implication is significant, as 42 signifies the end of tribulation (i.e., the Messiah has come). Furthermore, the names remind the reader of the continuity with the Old Testament, which is important from a traditional Christian standpoint to refute groups that wanted to discard the Old Testament completely. It's why Matthew comes first in the New Testament, actually.

Not only that, but there are further implications. The reader is reminded of the original covenant with Abraham, of the Exodus, the anointing of David and the promise made that the Messiah would arise from his house, the Babylonian exile and the entire spiritual tradition of the Jews.

Furthermore, the whole concept of Hebrew tradition is stood on its head at the very end. The list begins like one of the numerous lists from the Old Testament, then ends by reminding us that Joseph was not Jesus's father. Spiritually, Jesus is the descendant of this house, but physically he is not. He fulfills the promise made, over and over again, to the forebears of Joseph but he comes directly from God.

Also, the unusual circumstances of his birth are legitimized by mention of the four other women prior to Mary who get referenced in that whole list of "begats". Tamar, who posed as a prostitute to have sex with her father-in-law, Rahab, the prostitute of Jericho who hid the Israelite spies before Joshua had the town destroyed, Ruth, who was a Moabite and not even Jewish and whose marriage to Boaz was very unconventional, and Bathsheba, who committed adultery with David while her husband Uriah was fighting in the army (and placed so as to die).

There's a reason why they're mentioned. Anyone reading that whole list and familiar with the stories sees how everything comes together perfectly. All those "begats" are incredibly important in tying the Old Testament to the New.