Well, Abstract Algebra is a fundamental tool with many applications for natural sciences like Chemistry (Quantum Orbital Theory, etc.) and Physics (Quantum Mechanics, etc.) but this answer will focus on why it is important for Mathematics itself.
We all know that algebra is the language of Mathematics. It is the essential tool we use to notate our Mathematical ideas. In this case we can say that Abstract Algebra is the Linguistics of Mathematics. Every set of mathematical objects (real or complex numbers, points on elliptic curves etc.) talks a different language. Some of them are commutative under our operations (i.e. the 'verbs'), and some of them aren't. Let me give a better example. Let's think the set of polynomials with the property,
$$p(x) = 0$$
for $x = 1$. If you play with these polynomials soon or later you will realize that this set is closed under addition and scalar multiplication. Yes! You discovered the Vector Spaces now! As soon as you start to study and investigate their general properties, you will find out that the language of Vector Spaces is not only talked by the polynomials but also many other like $R^N$, $C^N$, etc. Since you already know the language they talk, even if you do know nothing about them, you will be knowing everything about them. This is the whole point of Abstract Algebra and it is really useful!
For sake of another example: If you know the language of groups (AKA group theory among mathematicians), you won't need to verify Euler's Totient Theorem since we already know Lagrange's Theorem holds for any finite group (in this case the finite group $U(n)$ closed under whatever). We already know everything about our mathematical objects even if we do know nothing about them.
Also why it is called abstract? Well, simply because for being able to talk about languages we must do an abstraction over mathematical objects and look for common properties between them.
This is my interpretation of the subject.