[masaegu]

I did some research myself and there seem to be some theories regarding this. The one that appears to be most supported by linguists is as follows:

まらする originally had three usages:
1. Humble form of "to give". (present-day 差し上げる)
2. Humble form of "to say" or "to ask". 　(present-day 申し上げる and お～～いたす, respectively)
3. Polite affirmation. (present-day ます)

Around 12 - 16th century, Usage #1 disappeared and around 16 - 17th centuries, Usage #2 disappeared. Currently, only Usage #3 has remained though the word has changed its form to ます.

Not much info, I know, though. I'm completely new to this myself.