"Being responsible for" something means that you have to take care of it and make sure that it's OK. For example:

I was responsible for taking care of my little brother.

This means that it was your job to take care of your brother. If something bad happened to your brother, you would be blamed for it.

Here's an example that you might see on a sign in the locker room at a gym:

Total Fitness Club is not responsible for any missing or stolen items.

This means that the gym will not have to pay for your things if they get stolen.