I know my crypto wallet does not contain my actual coins but a private key. Is this all that is contained in a wallet? if so how is it possible to restore a wallet on another computer by just knowing the pass phrases?
Think of it as a mathematical equation of a curve with two value X and Y to represent a point on that curve.
Now crypto keys work the same way.
X is your public key (address) Y is your private key (secret)
All the funds of this wallet are written on the blockchain. To spend them you have to resolve the curve by providing Y to the algorithm as a proof of ownership.
This is basically how it works!
the 12 words seed phrase are just a convenient way of representing bytes so you can write them down but they are bytes.