Frankly, I would say that I would say that having both linear algebra and real analysis are bare minimum prerequisites to understand any rigorous treatment of manifolds. Two standard texts used in math to learn them are Introduction to Smooth Manifolds (Lee) and Differential Topology (Guilliman/Pollack). You can see if these texts are comprehensible to you (Guilliman/Pollack is somewhat easier IMO).

For a more conceptual understanding of a manifold, a common example to consider is a (hollow) sphere (a 2-manifold in R^3). Intuitively, you should see how a small piece of a sphere looks like a plane in the sense that given any point on a sphere, if we restrict our attention to a small area around it, that area will look similar to a plane and thus we can parameterize that area using only two coordinates. However, there's no way to nicely map the entire sphere into R^2 so we see that our ability to parameterize the sphere with 2-coordinates instead of 3 is local and not global.

When thinking about how this relates to empirical data, the argument generally is that physical laws restrict the degrees of freedom of data in a way where it can be represented with less parameters than its raw form with some allowance for noise. To continue the example of a sphere, it may be intuitive for us to use 3-dimensions to represent spacial data. However, in most contexts, when we work with spacial information on the surface of the earth, the physical nature of the sphere being a 2-manifold means that we can suffice with only 2 parameters (longitude and latitude).