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 an isomorphism (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

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