[nyhc99]

another way that word is commonly used is to indicate the person who has a responsibility; the one who is tasked with handling a situation. ie "who is responsible for cleaning up this mess?"

I take it that's what was meant by the notion that you're responsible for everything that happens to you. In this case it has nothing to do with praise or blame. Rather, it means that whether something happened to you by chance or through your own fault, the onus to clean up the mess falls upon you.

[gjm11]

Hmm, maybe. The way I'd say it is that dealing with whatever happens to you is generally your responsibility, not that the things that happen themselves are.

("Generally", not literally always. Extreme example: If you get hit by a car and your injuries put you in a coma, then both fixing the injuries and taking action against the driver if appropriate are necessarily other people's job.)