You don't really need modular arithmetic to explain it!

A sketch:

1 is 1 greater than 0, which is a multiple of 11

10 is 1 less than 11, which is a multiple of 11

100 is 1 greater than 99, which is a multiple of 11

1000 is 1 less than 1001, which is a multiple of 11

10,000 is 1 greater than 9999, a multiple of 11

(etc.)

Non-mathematicians seem to understand this explanation intuitively, particularly if you start with the even simpler one for 9 (left as an easy exercise for the reader).