I'll answer your questions one at a time.
He says that if we look at the sample ACF of the returns (The first
plot), there are no signs of significant serial correlation. However,
if we look at the ACF of the returns squared (The second plot), it
tells us that the return series is indeed serially uncorrelated, but
dependent. Could someone explain how he came to this conclusion?
On the first chart (of raw returns), the ACF does not appear to be significantly non-0 anywhere (other than at lag 1 of course). In other words, there is no serial correlation of the returns.
On the second chart (of squared returns), the ACF does appear to be significantly non-0 at certain lags. So this gives us hope that we can use these squared returns to predict something by using them.
(Interestingly, if you plotted the ACF for absolute returns, you would find something similar to the second chart. This is because the absolute value and the square both discard the sign to measure some sort of "deviation", in the non-technical sense of the word.)
And in general, what is the intuition behind looking at the square of
the series when we are searching for the "ARCH" effect?
It's not just any series, it's a series of raw returns. And as you saw in the first chart, raw returns are not serially correlated.
However, squared (and absolute) returns are, which is good news! This is because we can now use them to predict "volatility" (i.e. the conditional variance) using ARCH models. And this is what ARCH models do.
Incidentally, I wouldn't use the term "ARCH effect". What I would say is that ARCH-type models take us some way towards predicting the conditional variance of the returns of financial assets.
A good reference on this topic is Taylor.