What do the symbols in this image mean in relation to autoencoding? (which concepts do they represent and why?)

The question seems more about notation than autoencoders:

$ a,b = argmin_{a,b}f(a,b)$ simply means that $a$ and $b$ are chosen to minimize $f$.

w.r.t. autoencoders, the argmin statement is saying that we want the decoding of the encoded data $X$ to be as similar as possible to the original $X$.

To represent "decoding of the encoding", we can use function composition: $psicircphi X$ which means $psi(phi(X))$, hence the decoding of the encoding of $X$.

So overall, we are saying that we choose the encoding and decoding functions to minimize the delta between input $X$ and encoded/decoded $X$.

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Where the question asks:

what does the arg min section and beyond mean?

The latter half "and beyond" should be more specific.