So you have a function: $f(x,y) = xy$ that graphs some 3 dimensional surface. And for a reason, you need to change the input variables to be functions of $t$. So you let $x = x(t)$ and let $y = y(t)$. After that, you rewrite the function as $f(x(t),y(t)) = xy$. This is quite common to do from my point of view and it doesn't make sense to me.
In general, why do people do this kind of substitution and what is it for? What is the input $t$ mean and how does it relate to the functions $x(t)$ and $y(t)$? How does the input $t$ map to the $x$ and $y$ values? I would like to understand this idea so I don't frown the next time I see this done.
One way to think of this is that you are looking at the values of $f(x,y)$ along some path in the $x-y$ plane. For example, maybe $f(x,y)$ represents the energy of a particle in space, and you know that the particle is moving in a circle. You can describe its path by the parametric equations $x(t)=\sin(t),\ y(t)=\cos(t)$, where $t$ represents time. Then $f(x(t),y(t)$ would give the energy of this particle at time t.
The advantage of this approach is that you can provide the value of the energy for arbitrary paths, and you can write it's time derivative in terms of the partial derivatives of $f$, etc. So it simplifies some calculations and allows you to make arguments in greater generalities.
I'm sure people will provide some other ways of thinking about this, but this is what jumped to my mind.