Is Abstract Algebra effectively Pre-Calc Algebra but more "abstract"? I have taken up to Pre-Calc/Algebra, and being made to plug and chug in a ready-made formula to arrive at an answer is boring. 
I would really love to delve into the "why's" behind why certain formulas and algebraic techniques (such as completing the square or matrices/Kramer's rule) work in certain scenarios and why they break down in others. 
Is this what Abstract Algebra is all about? Answering the "why" behind why we do certain operations/computations in PreCalc-Algebra? 
I want to get creative with my own intuitive approach towards solving problems and not rely on regurgitation of formulas I don't fully understand from where they came.
 A: I do not think "answering the why's" is the key difference. Mostly, in abstract algebra you'll be dealing with two things not present in PreCalc Algebra: proofs and argumentation. A problem in PreCalc may be: "say if X satisfies this property" or "Find the Y of this Z". On the other hand, Abstract Algebra will come along and say: "Suppose B. Prove that N is then M", and that might not be a trivial thing to even think about.
Personally, when I took Abstract Algebra I felt the main difference was learning to understand and construct mathematical (rigorous) arguments, rather than memorizing results and applying on a need-to-know basis. It won't be plug-and-chug, and it won't be "This is why that is so". It'll be "Show rigorously that you can go from A to B" (which, if you have the imagination to interpret results, is a fantastic way of learning).
Read theorems and understand them. Then when you think you have them in your pocket, read them again. Know what you can and can't do, what can and can't happen, and you'll see just how beautiful (and kind of black-magic easy) Abstract Algebra can be.
A: Not relying on regurgitation of formulas you don't understand is what mathematics is all about. What you're maybe concerned about is inductive vs. deductive reasoning, where instead of getting a formula and a set of problems for the formula to help you realize that the formula really works (which would be an example of an inductive conclusion, aka "it really works for these 1000 examples, it must work for all examples"), you want to arrive at a conclusion from a set of hypotheses based on certain logical rules (which is deductive reasoning).
Inductive reasoning is used in early mathematical education, as deduction and proofs etc. are too abstract for young kids and early teens. As you progress through your education, (depending on where you're from, either in later high school or university) your courses should start being more deductive in their nature. A great example of this is Euclidean geometry - the ancient Greeks were one of the first civilizations to achieve a somewhat formal treatment of a mathematical subject, and learning from their approach in establishing geometry is often a great starting point for one's deductive "career"!
To answer your question about abstract algebra, let's say very crudely that it studies ordered $(n+1)$-tuples of the form $(S, \cdot_{1}, ..., \cdot_{n})$, where $S$ is a set, and $\cdot_{k}, 1 \leq k \leq n$ are operations, i.e. mappings from $S^{i_{k}}$ to $S$, where $i_{k}$ would be the length of the operation (the number of arguments the operation takes - for example, $+$ and $\cdot$, addition and multiplication of natural numbers, are binary operations - they take two arguments). Obviously, abstract algebra also studies many, many more things; drawing boundaries between areas of mathematics has always been difficult, and is especially difficult today. The difference between abstract algebra and algebra taught in early high school would almost definitely be the approach - abstract algebra takes a strict mathematical approach, and all reasoning done is deductive (although inductive reasoning is an extremely useful mathematical skill, as you can often get abstract ideas by looking at concrete examples - examples inspire questions and curiosity, which are two of some of the most important things in mathematics).
