The Christians stole the winter solstice holiday off the Pagans and renamed it.
This theory is pretty dubious in light of modern evidence. The earliest source that links the 2 only appears a millennium later.
Given that early Christians went out of their way to differentiate themselves from Pagans, and that it is likely a blatant attempt to paganise Christianity would have proved controversial, we could expect some sources noting dissident Christian groups had this actually happened. This is especially true if it were a top-down attempt by some Emperor or another.
It also doesn't explain the difference between when Eastern and Western Churches celebrate Christmas (and Easter). Dating Christmas from Easter/the Annunciation does cover this and has a stronger historical and theological grounding. Easter developed from the Christianisation of Passover.
It is far more likely that the holidays simply coincided, and so numerous pagan traditions became part of the Christian festivities as populations converted.
So Christmas festivities borrowed much from the pagans, but there was no calculated decision to 'steal and rename' the winter solstice by pretending it was when Jesus was born.