Let me try to write an informal explanation as to why (and why not) you might have $f_* \mathcal{O}_X = \mathcal{O}_Y$.  This is basically what J.C. Ottem wrote, but I'm trying to explain the reason at a slightly more philosophical level.

Now $O_X$ is the sheaf of regular functions on $X$.  Given an open set $U \subseteq Y$, the sections $\Gamma(U, f_* \mathcal{O}_X)$ is just $\Gamma(f^{-1}(U), \mathcal{O}_X)$.  For this to be viewed as even a subset of functions on $U$, you would expect it to be constant / well-defined at the points of $U$.  So consider some (closed) point $z \in U$.  Therefore, you need a section $\sigma \in \Gamma(f^{-1}(U), \mathcal{O}_X)$ to be constant on the fiber $f^{-1}(z)$.  Since $f$ is proper, this fiber is also proper, and thus the only sections are constant.  I just lied of course, the only sections are the functions that are constant on each *connected component* of the fiber.

Thus if you have fibers with multiple connected components, then you will expect that some of the sections $\sigma$ might be able to distinguish those connected components, and thus those sections of $f_* \mathcal{O}_X$ can't be viewed as functions on $Y$.

Why does normality come into play?  Well, the picture isn't quite as simple as what I just described.  If a scheme $Z$ is non-normal, and its normalization $Z' \to Z$ is injective/bijective (for example, the normalization of the cusp), then you should view that normalization map as the inclusion of all the ``algebraic functions'' which can be defined on the points.

In fact, given any scheme $Z$ over an algebraically closed field of characteristic zero, the *seminormalization* $Z'$ of $Z$ can be exactly described as ``the scheme whose structure sheaf has all functions that make sense on the closed points of $Z$.''

This is the point of view on seminormalization is described in:
Leahy and Vitulli, *Seminormal rings and weakly normal varieties.*  Nagoya Math. J.  82  (1981), 27–56