As for why you should care, depends on how you view continuity of identity.

Before you get to exotic* applications of quantum mechanics, you should cover the basics. Quantum mechanics forces a view of personal identity that treats the many words equally. This is true even if you endorse collapse, as long as people can occasionally be put in superposition.

In a classical world with randomness, you could imagine that there are lots of parallel copies of people, each with a serial number ("indexical uncertainty") who march in parallel until random events occur whose outcome is determined by the serial number. In such a setting, you could imagine that the invisible serial number is a core part of the person's consciousness and there is no continuity of identity between them. But QM says that quantum sources of randomness are not like that. It is not meaningful to talk of which parallel person gets which result.

* a euphemism for "wrong."