Real meaning of "beg to differ"

Yesterday I learned the real meaning of "begging the question." There's another phrase where you do the begging, and that is when you want to differ.

But I had a hard time trying to find out the real meaning and origin of the phrase, and went back to the site where I looked up the meaning of "beg the question." This was one the paragraphs:

This usage is understandable and has presumably come about by interpreting the 'beg' of 'beg the question' as 'request' or 'humbly submit'. This is the meaning of the word in the similar phrase 'beg to differ'.

I didn't know what that meant at first, but I quickly realized, through some googling, that it literally means what we think it does. To beg to differ means to politely disagree with someone.