Reddit mentions of Linear Algebra and Its Applications, Books a la Carte edition Plus NEW MyMathLab with Pearson eText -- Access Card Package (4th Edition)

I've structured my answer in 3 parts: how (abstract), why and how (now, concrete).

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How do I math?

Practice. Math is a skill that you learn. You learn math much like you learn a second language, or the way artists get better at shadowing. Some people are fascinated by mathematics in and of itself, but you can just as well consider it a tool. In either case, fluency is the result of practice.

Obviously I'm not talking about learning equations by rote, the way you might learn a foreign vocabulary. Practice is perseverance. You're flexing your brain in a way that it's not used to, and that's tiring. Some people think they're bad at math because they're staring in a daze at a set of equations, with a slowly building headache. This doesn't mean that "math isn't for you," or any such bullshit. You should liken it to the way your muscles start hurting when you're working out. Hell, consider it a rite of passage. Because that's the good news: it gets better. You can get fluent in handling the tools you've struggled to learn, and each new tool exponentially increases the number of problems you're able to solve.

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Why do I math? And why do you math so strange?

Each new tool exponentially increases the number of problems you're able to solve. This expresses itself in a number of different ways:

1. New problem classes. Some systems are non-linear and require differential equations. Some systems don't have exact solutions and require numerical mathematics. etc. These all require special tools.
   
   
2. Expressivity. With a bigger tool box, it becomes easier to express properties about a system, or to capture information in an abstract model. Imagine if you had to write a dissertation using only the 1000 most common words in the English language (similar to XKCD's project). Hard, huh? You're basically moving some of the work up-front; struggling to learn these new concepts now makes it easier to communicate later on.
   
   
3. Alternative paradigms. This is the least tangible, but one of the most important ones. This also ties into the roundabout methods they sometimes use. You can often attack any abstract problem from many different angles. These paradigm shifts are something that also only come from lots of practice. You're 100% right that there's no immediate gain in solving a problem in a roundabout way, but...
   
   a. Flexibility. You won't always see, or be able to solve those problems in a straightforward way. One of my high-school teachers once told me that "[I] went from Boston to New York via Paris". I told her she was right (I couldn't see how to solve the problem, so I took a bit of a detour), but that I did end up in New York.
   
   b. Easier to expand. There's a difference between solving a problem (in which case you want the most efficient technique) or understanding how a problem might be solved (what solution strategies exist and most crucially why they work/"the idea"). By being able to look at a problem in different ways, you gain additional understanding. You might be looking at a new mathematical concept and say, "hey, this is really just X in a different way."
   
   tl;dr: Math is many tools. You won't always have the best one for the job. You need to build a framework of understanding: you'll want to see the relation between different solution strategies and why they work. By knowing these relations, you'll be able to attack problems from multiple fronts. This ability makes you a better scientist.
   
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   So HOW do I math?
   

   
   Your question shows you have the will. What you're looking for is a way. Not knowing your background, I don't know how good you are at learning stuff, at what level you're working (HS/Uni), etc. but I can give you a few tips that come from my own experience. Your mileage may vary; I'm no authority on education.
   
   

4. It doesn't matter how you learn. Most programs require you to pay attention to a lecturer. Personally, I suck at this. Maybe you do to. It doesn't matter! If you're better at getting your information from books, read^(†)! If you're better with videos, watch^(††)! Find the way that works for you. For me, it's with a book, alone in my room. I don't go to any lectures anymore, but my grades are great! Other people I know are exactly the other way around. There's no one-size-fits-all.
   
   
5. Practice. Yes, we went over this, but seriously. Try to do theory + exercises in parallel. It cements your understanding and makes math more fun (once you reach a certain fluency, solving new problems isn't much different from solving, say, sudokus).
   
   †: Look to the reference material of your course first. If you're at HS level I have no particular recommendations for you, but you could definitely get your fix from one of the math or teaching subreddits. For Uni level, Pearson books are really good for most sciences. I'm a big fan in particular (non-Pearson) of Lay's Linear Algebra and its Applications for (you guessed it!) linear algebra.
   
   ††: Kahn's academy is great for HS level. MIT open courseware (or the equivalent thereof for Harvard, etc.) is great for uni-level stuff. Being able to change the playback speed, pause & rewind makes all the difference for me.