Stalemate/Illegal moves

Hey everyone!
Newbie here 🙋‍♂️ I'm doing the introductory lessons (I literally knew nothing about chess until 5 days ago), and I reached the Stalemate lesson. And, well... I just don't get it 😅

The first example in the lesson is this one: 

White to move - White Qg8 - Black Qf6 - Black Kh7

As the white King is not in check, and doesn't have any legal moves to make, it's a stalemate. This may be an extremely dumb question for me to ask, but: how in the hell are f8, f7, g7, h7 and h8 illegal moves? The white King should be able to move to any of those squares... why is it not allowed?!

Same in the next example:

Black Ka1 - White Qc2 - White Kc3

Black is cornered, but why are a2, b2 and b1 not legal squares for Black to move to?
The first lessons teach you how to move your pieces, and the basic rules of how a check works and stuff... but they don't explain why a move is illegal, so I'm kinda lost.

Again, I'm sure all of this is pretty obvious for someone who isn't new to this, but for someone who is just strating to learn the chess rules like me, this stalemate thingy sounds like nonsense. You cornered your opponent, and for some stupid rule the game ends in a draw? That... kinda sucks 😅

So, yeah... I hope someone can help me clarify this, cause I ain't getting it.
But I'm really looking forward to learn more about this amazing game tho ♟
Cheers, everyone! 🎩

You are not allowed to put your King in a check.

Ah... that makes sense lol 😅 I apologize for the dumb question. Like I said, really new to the game, and didn't realize that tiny little detail 🤦‍♂️ Thanks for the help Toldsted 🤜🤛