Here's the thing - "what's" is one of those pesky contractions with more than one meaning. In the case of the phrase your teacher said, "what's that mean?", he or she is using it as "what does that mean?" However, it also means "what is". If someone said "what's that?" or "what's up?" ( as a colloquialism/slang/idiom for hello or how is everything), they would be using it as "what is that" or "what is up". So, the shorter answer: it means both "what is" and "what does", it just depends on the context. (It can also mean "what has", but that usage is less common, at least in American English.)

As Roland mentioned, it's American everyday speech. But it is not grammatically correct, since "does" never is abbreviated to " 's "

It's grammatically correct as far as spoken English goes. In relaxed speech, 'What does ...?' is sometimes contracted to 'What's'. The underlying grammar, if you like, when a native speaker says 'What's that mean?' is 'What does that mean?'. So it's not a mistake. However, in written English, we don't use this contraction. In written English, only 'is' and 'has' can be contracted to 's'. I realise that's a little confusing and/or illogical, but I hope it makes sense.

What's (what does) that mean? I can't tell you that it is grammatically correct, but it is a very American everyday speech.