Should one learn proofs for the correctness of algorithms? Why?
That's up to you. Reading correctness proofs for algorithms is useful since:


The proofs explain why the algorithms work. If you need to modify these algorithms, you'd better know how they work. Such an understanding is very useful even just for coding algorithms, and certainly for debugging them.
The proofs are good examples of mathematical proofs. The mathematical notion of proof is one of the most important contributions of mathematics to the world, and is a way of knowing which you could find useful in life, even without using it yourself.
If you continue in academia, say get a graduate degree, you might have to write such proofs in the future.


However, it is my understanding that most programmers don't come up with new algorithms, or even code classical ones. If this is your case, you can probably skip the proofs, indeed the entire course, if it were not made compulsory by your university (if any).