Why is it so bad to mock classes?

IMHO, what your colleague means is that you should program to an interface, not an implementation. If you find yourself mocking classes too often, it's a sign you broke the previous principle when designing your architecture.

Mocking is used for protocol testing - it tests how you'll use an API, and how you'll react when the API reacts accordingly.

Ideally (in many cases at least), that API should be specified as an interface rather than a class - an interface defines a protocol, a class defines at least part of an implementation.

On a practical note, mocking frameworks tend to have limitations around mocking classes.

In my experience, mocking is somewhat overused - often you're not really interested in the exact interaction, you really want a stub... but mocking framework can be used to create stubs, and you fall into the trap of creating brittle tests by mocking instead of stubbing. It's a hard balance to get right though.