In other words, why is it 10 or 11 days after the winter solstice? What is special about the day (out of all 365.25) to be January 1st? Is it just that it is "midwinter"?
According to wikipedia's page on the Julian Calendar:
The first step of the reform was to realign the start of the calendar year (1 January) to the tropical year by making 46 BC (708 AUC) 445 days long, compensating for the intercalations which had been missed during Caesar's pontificate.
Wouldn't aligning it to the "tropical year" mean January 1st on the Winter Solstice?
I'm not asking why the year starts on January 1, that seems to be a complicated and convoluted topic.
Check out the FAQ section on calendars. This thread in particular seems to be the one you want:
Why does the new year start 10 days after the solstice? by /u/aescolanus
But the other ones might be of interest as well.
TL;DR it was the Romans.