Before we talk about how mathematicians think about four dimensions, let's talk about how mathematicians think about two dimensions.
Here's a point! You've probably seen these things before. Maybe it represents a point on a map, or where on Earth you live. But this dot alone doesn't tell us anything unless we know how to describe the dot's position.

How do mathematicians describe points in 2D space? The most common answer, which you've probably seen before, is an x-y coordinate system!
This coordinate system describes positions using a list of two numbers, and each number represents how far the point is along one axis. Pretty simple.

Here's a different way to think about an xy coordinate system: You can think of x and y as points on two different number lines. Then, to get the final position on the plane, you can recombine the two axes together.
An xy coordinate system has some nice properties - different lists of coordinates give you different points on the plane, and every point can be reached by some list of coordinates. Keep this in the back of your head - it'll be useful soon.

How do mathematicians think about 3D coordinates? The exact same way, but instead of a list of just two x and y coordinates, we use three, called x and y, and z.

Just like in 2D, we can think about this 3D xyz coordinate system as either a list of three numbers, or as three independent number lines, where all possible combinations of points on lines are recombined to cover every possible point in 3D.