Why does the autocorrelation get its peak at zero?
Are you looking for a formal proof or the intuition behind this? In the later case: "Nothing can be more similar to a function than itself". Autocorrelation at lag $\tau$ measures the similarity between a function $f$ and the same function shifted by $\tau$. Note that if $f$ is periodic, $f$ shifted by any integer multiple of $\tau$ and $f$ coincide, so the autocorrelation has a comb shape - with peaks at the integer multiples of the period with the same height as the central peak.