The simple and obvious answer: they come from the Romans. But the whole story on how the months of the year got their names is a little bit more interesting and includes bits about how the calendar first started in March, how they just started naming months after numbers after June, how there were month-less days, and how Julius Caesar tweaked it all. History is fun.
It does bug the hell out of me that our 9th, 10th, 11th, and 12th months are named Seventh-month, Eighth-month, Ninth-month, and Tenth-month. I mean our days of the week all have real names; they didn’t get to Friday, decide to knockoff for and early weekend, and just call the last two days of the week Quinqueday and Sexday (though I suspect people would be cool with Sexday).