begged to decline

Can anyone help me about the exact meaning of the words in bold?

Jack's embassy contact, John, begged to decline.

I shouls add that jack proposed an idea that John could not accept it.

Solution 1:

It means he very very politely refused - or possibly, that he had no intention of accepting the suggestion, but humorously pretended to be very polite about it.

"Decline" here means "refuse an invitiation". "Beg to decline" is a (now very old-fashioned) way of softening this for politeness' sake: the implication is that you're very apologetic about declining, so you are begging the other person to allow you to decline.