Let $f:X\to Y$ be a continuous map between the topological space.
Let $\mathcal{F}$ be a sheaf on $X$, then we can push forward it to $Y$. Let $s\in \Gamma(U,f_* \mathcal{F})$ be the section of the direct image sheaf on $U$, since $\Gamma(U,f_* \mathcal{F}) = \Gamma(f^{-1}U, \mathcal{F})$ , $s$ can also be treated as section of $\mathcal{F}$ on $f^{-1}(U)$ denote it $\tilde{s}\in \Gamma(f^{-1}U, \mathcal{F})$.
Define the support be $ \text{supp }s =\{x\in U\mid s_x \ne 0\}$ ,we have also $\text{supp }\tilde{s} = \{x\in f^{-1}(U) \mid \tilde{s}_x \ne 0\}$.
What's the relation between $\text{supp }\tilde{s} $ and $\text{supp }s$, and how to argue it?
The behaviour of stalks of $f_\ast\mathcal{F}$ can be a little tricky to wrap your head around, since the topology on $Y$ (in particular, the family of open neighbourhoods of a point $y \in Y$) can potentially lack dependence on the topology on $X$.
For instance, if $Y = \{\ast\}$ is just a point, then the pushforward of a non-zero global section $s \in \Gamma(X,\mathcal{F})$ to $Y$ is of course going to have $\{\ast\}$ as its support, whereas $\text{supp}(s) \subseteq X$ might be a proper subset in $X$. As this example illustrates, we evidently we always have $\text{supp}(s) \subseteq f^{-1}(\text{supp}(f_\ast s))$, since the collection of open neighbourhoods containing a point $x \in \text{supp}(s)$ contains its sub-family given by the collection of open sets $\{f^{-1}(U) \mid f(x) \in U, \text{ open in }Y\}$.
If $X \subseteq Y$ is a subset and $f$ is just the inclusion map, then your two considered subsets coincide by definition of the topology induced on $X$ by that of $Y$.