There's a lot one could say, but I'll try to be brief. Roughly the idea (just like with the zeta functions) is that L-functions provide a way to analytically study arithmetic objects. Specifically a lot of interesting data is encoded in the location of the zeroes and poles of L-functions, and because L-functions are analytic objects, you can now use analysis to study arithmetic. Here are some examples:

The fact that $\zeta(s)$ has a pole at $s=1$ implies the infinitude of primes.

**(added)** The Riemann hypotheses and generalizations, which are about location of nontrivial zeroes of zeta-/L-functions, have lots of implications, such as refined information about distribution of prime numbers.

The fact that Dirichlet L-functions do *not* have a zero at $s=1$ implies there are infinitely many primes in arithmetic progressions. Dirichlet introduced the notion of L-functions to prove this fact.

If $E : y^2 = x^3+ax+b$ is an elliptic curve and its $L$-function $L(s,E)$ (which is also the $L$-function of an elliptic curve) is nonzero at the central value $s=1$, then $y^2=x^3+ax+b$ has only finitely many rational solutions. This is the known direction of the Birch and Swinnerton-Dyer conjecture.

**(added)** In addition to knowing just locations of zeroes and poles of L-functions, the actual values of L-functions at special points contain further arithmetic information. For instance, if $\chi_K$ is the quadratic Dirichlet character associated to an imaginary quadratic field $K$, then the class number formula says $L(1,\chi_K)$ is essentially the class number of $K$. Similarly, the value of $L(1,E)$ in the previous example is conjecturally expressed in terms of the size of the Tate-Shafarevich group of $E$ and the number of rational points on $E$.

As mentioned in the comments, $L$-functions are also a convenient tool to associate different kinds of objects to each other, e.g., elliptic curves and modular forms, but are not strictly needed to do this.

Nice $L$-functions will have at least meromorphic continuation to $\mathbb C$, Euler products, and certain bounds on their growth. For instance, L-functions of eigencusp forms and Dirichlet L-functions. These properties make $L$-functions nice analytic objects to work with. In particular, the Euler product provides a way to study global objects from local data (one finite set of data for each prime number $p$).

**(added)** See also this MathOverflow question.