How does AES-256 works if AES is only a 128-bit block cipher?

I'm currently studying AES encryption and I'm quite confused on how do we use a 256-bit key in AES?

Becuase in my understanding, AES no matter what the key size is, it still operates and encrypt a 128-bit block of message only in the whole cycle/rounds until it reaches the end.

well for AES-128 we use a 128-bit key that is total of 16 byte char, and that can be represented by a 4x4 matrix, and most AES operations and steps works on 4x4 matrix right?

Then how do we use a 256 bit key?, that is a 32 byte char and that is not even a perfect square right?

Will that 256-bit key result into the same size of 128-bit key version of AES when expanded by key expansion? and just use 16 bytes each when encrypting to match the 4x4 state array?

If you're wondering why a 256-bit key offers any greater security, you need to realize that a 128-bit block cipher is essentially a way of mapping $2^{128}$ input values to $2^{128}$ output values. That means the set of all possible 128-bit block ciphers has a size of $2^{128}!$, which is about $10^{10^{40}}$.

A 128-bit key allows you to choose one of $2^{128}$ possible mappings. Although this is a huge number, it's a vanishingly small fraction of the set of all ciphers. The same could be said for any practical key size, but with a 256-bit key, you get to choose from a far larger set of possible input-output mappings, making the encryption that much harder to crack.