What does the size of the evaluation bar mean?

For example, if the bar is 70% white and 30% black, what exactly does this mean? I know white is winning, but does the color ratio have a mathematical meaning?

I'm not talking about the number written on the bar.

Number and the size of white and black part of the bar are directly proportional (if I may call it like that).

So When it says M1 for white, the entire bar will be white for instance, and when it says +0.00 the bars will be equal.

Hopefully that answers the question. 

It doesn't work like that. Those bars are just graphic representations of numbers shown in the evaluation. Engine doesn't work with probabilities for a win. It just works with estimates represented with pawns advantage or, if the engine can calculate checkmate, it directly shows in how many moves will the checkmate occur.

Is there a mathematical function that converts evaluation number to bar size?

The amount of black or white in the evaluation bar represents the positional balance according to Stockfish. Stockfish can see a forced checkmate if the entire bar is one color.