In my original post, (preserved below) I had forgotten that Riemann surface is the name given specifically to complex manifolds of 1-dimension. These are two dimensional real Riemannian manifolds, but with added structure. Further, their most common use is as the surfaces generated by analytic continuation, which is what I think the OP is really asking about.
These surfaces are generated by plotting holomorphic functions $w = f(z)$, and then extending them by analytic continuation. So we are looking locally at the graph in $\Bbb C^2 \equiv \Bbb R^4$ of a function $w = f(z)$.
We can't visualize $\Bbb R^4$, so instead we plot in $\Bbb R^3$, using the horizontal plane $x_3 = 0$ as the plane of the complex $z$, and the vertical axis to plot the real part of $w$. However, we still need to represent the imaginary part of $w$, and for that we turn to color. We allow the color spectrum to represent the imaginary axis. (You might complain that the color spectrum is bounded. But we cannot plot more than a bounded range of any graph, so this doesn't restrict us any more than usual.)
On the Wikipedia page for Riemann surfaces, the first image is the graph of $w = \sqrt z$, and in the caption, it mentions that the height in graph is the real part of $w$. But, alas, rather than explain that the coloration represents the imaginary part of $w$, it instead mentions that rotating the graph by $180^\circ$ will give the imaginary part, a trick that works only for this particular function.
Orignal post, treating "Riemann surface" as meaning two dimensional real Riemannian manifold:
Riemann surfaces are just abstract 2-dimensional surfaces. They do not in and of themselves have "height" or "color" as attributes. A Riemann surface is not even a surface in space. It is just an abstract object.
However, we can only visualize things in space, so we generally think of these surfaces as existing in the 3-dimensional space that we are so familiar with. Such a placement in space is called an "embedding" of the surface. The embedding is not an attribute of the surface itself, but something we add to it so that we can visualize it, or make use of it in relation to other objects in space. It is only by this embedding that a Riemann surface gains "height", meaning its $z$ coordinate.
For example, $S^1$ is the name we give to the circle. The cross-product of this circle with itself, $S^1 \times S^1$ is a Riemann surface. There is complete symmetry between its two dimensions. They behave completely the same. But $S^1$ can only be embedded in $\Bbb R^2$ or higher. So the natural space to embedd $S^1 \times S^1$ is $\Bbb R^2 \times \Bbb R^2 = \Bbb R^4$. But we can only imagine 3-dimensions, so this natural embedding is not one we use. Instead we stretch one of the circles, making it much bigger than the other. Then we can fit $S^1 \times S^1$ into $\Bbb R^3$ as the torus, or donut shape. But only at the expense of losing the natural equivalence of its two coordinates.
It gets worse for other surfaces, such as the projective plane, or the Klein bottle. These surfaces cannot be embedded into 3-dimensional space, no matter what we do. No matter how we try to deform the embedded surface, it will always intersect itself, which violates the conditions to be a true representation of the surface. points that should be far distant from each other are instead close neighbors, or even represented by the same point. If you've seen Klein bottles, you will have noticed that at some point, its tube will pass through its own wall. This intersection is not part of the actual surface, but merely a compromise we had to make to be able to see it at all.
So to properly see these surfaces, we need 4 dimensions. But we can only see three. If only there were some way of depicting that 4th dimension. A common substitute for a 4th dimension is to use a color spectrum. The color value of a point, ranging from red to violet, is the 4th coordinate of an embedding of the surface into $\Bbb R^4$. In these visualizations, color tells you when two points that are close together in 3-D space are actually far apart.
I've seen a number of these, but am having trouble finding them online. So far the best I've found is this youtube video, where someone is clicking options in modelling software showing a projective plane embedding, and occasionally turns this dimensional coloration on. (The coloration the rest of the time is just for shading so that the two dimensional picture looks like a 3D object.) https://www.youtube.com/watch?v=yUerROXAEtw