“How is that my responsibility?”

English Lesson: How is that my responsibility?

Your wife is angry because she woke up late to go to work. She gets angry at you because you didn't wake her up.

How is that my responsibility?

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How is that my (something)?

English speakers sometimes say things like:'

How is that my fault?

This means "I don't think it's my fault." You say this when someone else blames you for something. Some other similar phrases are:

How is that my responsibility?

How is that my problem?

People usually sound angry when they use "How is that my ___?" 

(something) is (someone's) responsibility

"Being responsible for" something means that you will be rewarded if it's successful, and blamed if it fails.

If something is your responsibility, it means that it's your job to make sure that it's OK. It's your fault if something goes wrong.

We talk about responsibility:

Raising a child is the parents' responsibility, not the school's.

When I was growing up on the farm, it was my responsiblity to feed the chickens.