How do extension methods work under-the-hood?

Question

A contractor where I work is using extension methods to implement CRUD on well-known internal classes that we own. I say it is better

Answer 1

I assume extension methods make heavy use of reflection (which is slower).

No. Extension methods are resolved at compile-time, no reflection required.
That negates your performance concerns.

Is it better to use inheretance or extension methods ?

I would say neither. Use a Repository (DAL). An entity should be persistence-agnostic (so: no inheritance from a base that does CRUD) and not pretend to be involved where it's not (no extensions).

You are right that "Using extension methods obfuscates & confuses the source of the CRUD methods" but inheritance is not the solution.