I'm a high school student, so I have no idea what a Jacobian or a manifold is, but as someone who's self-studied linear algebra and abstract algebra, I think it's pretty complex and takes a rather smart/dedicated person to pass these classes, so you're definitely smart enough to understand these concepts.
In my opinion, these fields are kind of intuitive in some ways. Honestly, I have no idea how mathematicians came up with all of this -- especially the more advanced (well, advanced to me) parts of these fields like field arithmetic or Galois theory or all of this spectral theorem and Schur decomposition/lemma stuff that I'm learning about right now. However, I've seen a lot of good online explanations, especially from looking at answers at this site, so I think I have a better intuition than some college students. If you're a college student on a strict schedule, you don't have the free time to explore maths like I do, so my guess is that it's b) and that the education system doesn't focus enough attention on comprehension and instead focuses on getting students passing grades and degrees.
Now, I know the beginning of linear algebra quite well since I've reviewed and cemented those concepts and I can definitely tell you how matrices have helped me. Maybe this and some of the links below will help you gain intuition on some things.
Matrices help us solve problems of systems of equations. For example:
Using Algebra I knowledge, we can solve this using elimination. Divide the first equation by $3$, subtract the second equation by $4$ times that to get rid of $x$, divide the second equation by $\frac 7 3$ to solve for $y$ and subtract the first equation by the $\frac 2 3$ the secone equation to solve for $x$.
This is elimination, but this is the same process used for RREFing the following matrix:
3 & 2 & 1 \\
4 & 5 & 2
If you RREF that matrix manually, you'll basically end up with the same row operations as we manipulated the equations. Thus, matrices and RREFing them is literally just solving systems of equations with elimination. However, it's hard to see that because there's no variables; we're just manipulating coefficients. It's even harder to see that with 3x3 and 4x4 matrices when this becomes increasingly more time-lengthy and when some people would rather use substitution or guess and check for such systems. However, with this method of matrix and RREFing, we have an algorithm that makes it much easier to do this without thinking or with a computer. By using this repetitive, boring algorithm, I bet mathematicians were able to solve systems a lot faster since they didn't need to keep track of variables and they didn't need to choose between elimination or substitution. They could just do the algorithm out and there's no thought involved. By using this kind of repetitive/familiar format of matrices, it makes solving systems of equations faster. RREFing basically solves the problem of linear systems, so using this rote method matrices and RREFing allows mathematicians to solve linear systems quickly and focus on more complex, interesting problems.
Now, hopefully, this explanation helped you understand matrices, but there are some more advanced concepts that I think other explanations can help you with best:
- Least squares approximation $\rightarrow$ This video taught me why in the world we need things like "left nullspace" and "orthogonal complements." These concepts fit into real world problems like least squares approximations and it's pretty cool how abstract concepts in linear algebra can come together in something useful like this.
- Eigenbases $\rightarrow$ This taught me how eigenbases can speed computations up and thus how eigenvectors can be useful if someone needs to apply a certain linear transformation repeatedly.
- Abstract Algebra: Theory and Applications $\rightarrow$ This online textbook shows a lot of the applications behind abstract algebra and gives us a motivation for doing it. It has a lot of practice problems and these proof problems took me hours and hours to solve, but I have a much better intuition about this field because of it.
In short, I've learned my intuition about mathematics because of good online explanations and a lot of practice problems. Explore mathematics, listen to explanations online, ask questions to your professors and possibly this forum and you will gain a better mathematical intuition about mathematical concepts that seem complex to you now, but down the road, will make much more sense. One year ago, all of the concepts I've explained above (except maybe matrices) made no sense to me, but now, I've gained a much better understanding of these concepts, even though I know I have a long way to go and I think that's probably the place you are in. In a few years, the concepts that seem bizarre and odd to us now will hopefully make a lot more sense to us then, at which point we might be struggling with even more complicated mathematical concepts. In any case, good luck and I hope all of this helped!