Why study Lie algebras?

Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this. By differentiating the Lie group action, you get a Lie algebra action, which is a linearization of the group action. As a linear object, a Lie algebra is often a lot easier to work with than working directly with the corresponding Lie group.

Whenever you do different kinds of differential geometry (Riemannian, Kahler, symplectic, etc.), there is always a Lie group and algebra lurking around either explicitly or implicitly.

It is possible to learn each particular specific geometry and work with the specific Lie group and algebra without learning anything about the general theory. However, it can be extremely useful to know the general theory and find common techniques that apply to different types of geometric structures.

Moreover, the general theory of Lie groups and algebras leads to a rich assortment of important explicit examples of geometric objects.

I consider Lie groups and algebras to be near or at the center of the mathematical universe and among the most important and useful mathematical objects I know. As far as I can tell, they play central roles in most other fields of mathematics and not just differential geometry.

ADDED: I have to say that I understand why this question needed to be asked. I don't think we introduce Lie groups and algebras properly to our students. They are missing from most if not all of the basic courses. Except for the orthogonal and possibly the unitary group, they are not mentioned much in differential geometry courses. They are too often introduced to students in a separate Lie group and algebra course, where everything is discussed too abstractly and too isolated from other subjects for my taste.

Here is a very fundamental way to create interesting Riemannian manifolds: Let $G$ be a semi-simple Lie group, let $K$ be its maximal compact subgroup, let $\Gamma$ be a discrete subgroup of $G$, and form $G / K.$ This quotient is called the symmetric space attached to $G$.

The Riemanian structure comes from an invariant metric on $G$, and so $G$ acts as isometries on $G/K$ by left translation.

If you consider the case $G = SL_2(\mathbb R)$, you get $SL_2(\mathbb R)/SO(2)$, which is naturally identified with the complex upper half-plane (on which $SL_2(\mathbb R)$ acts via Mobius transformations; note that the point $i$ is stablized precisely by $SO(2)$), which is also the hyperbolic plane. Other groups give higher dimensional hyperbolic spaces (e.g. $SL_2(\mathbb C)$ gives hyperbolic $3$-space), the Siegel upper half-spaces (from symplectic groups), complex balls, and many other well-known spaces.

If you now take a discrete subgroup $\Gamma$ of $G$, you can form the double quotient $\Gamma \backslash G /K$. These are some of the most celebrated Riemannian manifolds in mathematics. In the case of $SL_2(\mathbb R)$, we know via uniformization that all genus $\geq 2$ Riemann surfaces can be described in this way. In the case of $SL_2(\mathbb C)$ we get hyperbolic $3$-manifolds, from symplectic groups we get moduli spaces of abelian varieties, ... .

Now (as the preceding discussion hopefully makes clear), lots of these spaces are known by other names that don't involve Lie theory, and can be studied in a non-Lie-theoretic way. But the Lie-theoretic perspective provides a unifying, and frequently clarifying, point of view. For example, cohomological or function-theoretic invariants of these spaces can often be described and computed via Lie theoretic tools (e.g. via Lie algebra cohomology of certain unitary representations of the group $G$).

As a concluding remark, let me note that a general principle is that when certain symmetries are implict in a given context (e.g. $SL_2(R)$ being the group of hyperbolic isometries of the upper half-plane), it is good to explicitly bring them to the fore and take them into account. In geometry, the symmetry groups that appear (of a space, or perhaps of its universal cover) are very often Lie groups. And so a little knowledge of Lie theory can turn into a powerful tool for investigating a given geometric situation.

P.S. I should also note that the study of spaces $\Gamma \backslash G/K$ for certain $\Gamma$ (so-called congruence subgroups) is one of the basic topics of the Langlands program, and the function theory and cohomology of these spaces (especially their representation-theoretic structure) is conjectured to govern a vast amount of number theory. Trying to understand and work on these conjectures was my own motivation for learning Lie theory.

Lie's motivation for studying Lie groups and Lie algebras was the solution of differential equations. Lie algebras arise as the infinitesimal symmetries of differential equations, and in analogy with Galois' work on polynomial equations, understanding such symmetries can help understand the solutions of the equations.

I found a nice discussion of some of these ideas in

Olver, Peter J., Applications of Lie groups to differential equations., Graduate Texts in Mathematics. 107. New York: Springer-Verlag. xxviii, 513 p. (1993). ZBL0785.58003.