A hash is the output of an algorithm that converts any input into a character string of fixed length. The same input always gives the same hash, but the input cannot be computed back from the hash. Casinos almost always use SHA-256.
Why it is the load-bearing part of the provably fair scheme
The casino generates a server seed and shows you its hash before the game. Since the seed cannot be derived from the hash, you cannot know the result in advance. But since the same seed always gives the same hash, the casino cannot change the seed afterwards either — if it did, the hash of the revealed seed would not match the one it showed you before.
That is the core of the whole provably fair system: committing without disclosing.
How to check
Copy the server seed hash from the game settings before betting. After rotating the seed pair the casino reveals the seed in full. Run it through any SHA-256 calculator and compare the result — they must match character for character. A difference anywhere means something has been changed.
What a hash does NOT prove
Only that an individual round was not faked. It says nothing about the size of the house edge, whether the withdrawal arrives, or the fairness of the bonus terms. See server seed and client seed.