Freshly printed. When paper goes through a rotary printing press friction causes it to heat up. Therefore, if you grab the paper right off the press it is hot.
The phrase most often alludes to hot news, that is, striking or sensational news.
Newsprint used to be printed by a process called ‘hot metal printing,’ which involved molten lead being introduced into a mold to form the printing block. Although the term only really makes literal sense for printed items which use that process, it is by extension now also used figuratively to refer to anything that is fresh and newly made.
Hot off (or from) the press (or presses) didn’t originate as a phrase until the 20th century; for example, here is an advertisement in the New Jersey newspaper The Trenton Evening Times, July 1910:
“Just hot off the press and a strictly up-to-date cut price sheet of great value to housekeepers.”
The only thing hot off the press here this morning is that, having been to the pool and then OD’d on world news, I’m about to go have breakfast and prepare my mind for the world of writing, which is a much more peaceful place.