Originally the months were just numbered, starting with March which was the beginning of the campaigning season, the growing season, etc. This is why the months don't match up to our reckoning of the calendar, as you noticed. January and February didn't even exist in the original calendar until ~700 BCE, it having only 10 months previously. Over time, months were renamed to honor gods and a couple Caesars. March for Mars for obvious reasons, July for Julius Caesar because that was his birth month, August for Augustus because many of his triumphs were in that month, and so on.

More on Roman calendars and time reckoning can be found in JPVD Balsdon's Life and Leisure in Ancient Rome.

Follow up question: Sept means 7. Oct means 8. They are the 9th and 10th months respectively. Why?