In this phrase, “way” adds extra emphasis to “too (adjective)”. It makes the meaning stronger.

Whoa, that’s way too expensive.

I was way too tired to go out.

This phrase is extremely common in spoken English, particularly with teens and younger adults.  

You might also say “much too (adjective)” or, more formally, “far too (adjective)”:

I'm far too busy to take on any other responsibilities right now.