"Hang one's hat on" is an idiom which refers either to something you depend on or to something you believe in. For example mnemonic device it is something that you can put your hat on to remember new word or password.

To hang one's hat can mean a place you call home or To hang one's hat on means a skill or trait or whatever that you use or rely on For example, learning a new language is a skill you can hang your hat on. Meaning, if you know another language, you can use that language to your benefit.

Hi Cecile I believe this means to rely on something. It is an idiomatic expression.