I know very little about quantum computing but my understanding is that it will just be a much more powerful computer and therefore all the current key sizes that we use for cryptography will be rendered vulnerable to brute force attacks.
No; Quantum Computing is a different computing model; some problems which appear to be difficult for standard computers are known to be quite easy for a Quantum Computer; for other problems, they don't appear to become all that much easier for a Quantum Computer.
It turns out that the public key algorithms that we currently use in practice (RSA, DH, ECC) are the former - they are much easier to break on a Quantum Computer than they are known to be on a conventional one.
These "postquantum algorithms" are algorithms based on these latter problems; that is, ones where we don't know how a Quantum Computer has much advantage in solving; that is, breaking them is infeasible on both a conventional and a quantum computer.
Why not just use AES but with bigger key sizes?
Actually, AES-256 (that is, AES with 256 bit keys) are believed to be perfectly quantum safe; in fact, I would argue that AES-128 is also practically-speaking quantum safe (because the best known quantum algorithm against it, Grover's, takes too long to be practical).
However, that doesn't answer your question "why don't we use AES for everything?" Well, one thing that AES doesn't do is public key operations; for example, we don't know how to use AES in such a way that someone with the public key can encrypt, but only someone with the private key can decrypt. We are used to these public key operations, which makes things convenient, and just AES doesn't give us that.