Suppose we have a function representing a physical quantity that depends on both position and time. In other words, at any particular time the quantity has different values at different places, and the quantity also changes as time changes.
For example, consider f(t,x)=tsin(x). Notice that there are two input variables, t and x, and they are both put inside the brackets after the f.
Here t represents time and x represents position (one spatial dimension only so we only know the function along a line). Here's a picture of the function f(t,x) at t=0. We write the function f(0,x) against the vertical axis to show that this is the function f(t,x) when t is zero.
It is just zero everywhere! That's because f(t,x) is given by tsin(x), and if t=0 then tsin(x) is also zero, for all values of x.
Now let's see how f(t,x) has changed by the time t=1. Here is f(1,x):
It is the curve of sin(x), since if t=1, then tsin(x) is just sin(x).
Between the time t=0 and the time t=1, the function f(x,t) has certainly changed, at every point of x, except at x=0 and x=p where it is still zero. Why is it still zero at those two points?
That means that if you were sitting at a particular value of x (but not at x=0 or x=p) then you would see f changing with time. In fact as time went on you'd just see f getting bigger and bigger, since tsin(x) just grows as t grows.
Now consider the function at a particular time instead, say t=1 as in the picture above. Now imagine walking along the curve. So that's keeping t fixed, at t=1, and letting x change.
From the curve above you can see that you'd be walking uphill for some values of x and downhill for others. So, for a fixed value of t, the value of the function certainly changes as x changes.
Any function of two (or more) variables has this feature, you can look at how the function changes as you vary any one of the variables, while you keep the other variable(s) fixed.