(Short) exact sequences are incredible tools, that often enable us to get information about big things from information about smaller things.
Saying that $0\to A\to B\to C\to 0$ is short exact essentially means that $B$ is "built" from $A$ and $C$. This "building" process is not as easy and naive as just $(A,C)\mapsto A\oplus C$, but it still counts as a "decomposition".
More generally, from an exact sequence $A\to B\to C$, you can hope to recover information about $B$ from information about $A,C$.
Getting long exact sequences from short exact sequences is important because you're often more interested in the homology than in the chain complexes - that way getting information about the chain complexes ("they form a short exact sequence") allows you to recover information about their homology ("it forms a long exact sequence"), which is what you're interested in.
When a chain complex is exact, this is giving you immense information about it : to get the existence of an antecedent (that is, to solve an equation) you only have to compute a differential (so you get "there exists a solution to my equation" from "this computation gives me $0$", which is extremely powerful). If your complex is also exact in further degrees, you can get information about how many solutions your equation has etc.
Take for instance the de Rham complex of a manifold : if it's exact at $\Omega^k(M)\to \Omega^{k+1}(M)\to \Omega^{k+2}(M)$, this is telling you that a $k+1$-form $\omega$ is $d$ of something if and only if $d\omega = 0$ : clearly the latter is easier to check in general. For instance, in euclidean space, since the de Rham cohomology is $0$, this means that if you want to check whether a given vector field is a gradient, you only have to compute its divergence - this comes in handy in physics for instance.
Exact sequences are applied in lots of places outside homological algebra; in algebraic topology and algebraic geometry (where they can be used to compute invariants, such as (co)homology of spaces, or other more complicated objects - e.g. from the Mayer-Vietoris exact sequence, and homotopy-invariance, you can compute the singular homology of spheres, and thus distinguish them and prove the Brouwer fixed point theorem), but also in most of algebra (e.g. representation theory, where you can use them for many purposes : decompose some objects into smaller, easier to study objects, reduce problems to simpler ones, etc.), and some parts of differential geometry (where you often have some (co)homology theories ling around), some parts of analysis (where you have sheaves, and so sometimes encounter cohomology) etc. etc.
See here for examples from lots of places : https://mathoverflow.net/questions/363720/short-exact-sequences-every-mathematician-should-know
