How to make the interactive Schnorr identification protocol become non-interactive
The most common way to change an interactive Schnorr protocol into a non-interactive one is to rely on a random oracle. This is the so-called "Fiat-Shamir heuristic".

Notice that a transcript in your case would look like $(r, e, y)$, where $e$ is the challenge the verifier sends to the prover.
Now, you want to get rid of any interaction with the verifier to get that random challenge $e$, so what you need to do is to find a way to get rid of that interactive challenge. And there, you could have a random oracle giving you the challenge instead of the verifier.

In practice, this is done using a cryptographic hash function $H$ and so your transcript becomes $(r, H(r), y)$ or, as on Wikipedia: $(r, H(g,y,r), y)$. (It is important to include the public parameters in your Fiat-Shamir, to avoid problems such as weak Fiat-Shamir heuristics.)

Now, you can still verify that it works just like the interactive case, excepted we don't need to ask Victor, the verifier, to provide us with a random challenge anymore, since both the prover and the verifier can compute the challenge out of the rest of the data in a non-interactive way.