The initial interest in the zeros is their connection with the distribution of primes, which is often done via asymptotic statements about the prime counting function. In analytic number theory, it is standard fare to have an arithmetic function defined by a summation formula, and then modify it into a form that is easier to manipulate and obtain results for, in such a way that the asymptotic results about the modified function can be translated into results about the original function very easily. This is certainly the case with $\pi(x)$, which is why I bring this up. Most of the information that is relevant here can be found on the Explicit formula article at Wikipedia, for explicit formulas for the $\pi(x)$ function using the zeros of the Riemann zeta function. Two key highlights:

$(1)$ *"This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their 'expected' positions."*

$(2)$ *"Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors."*

With the very basics of complex numbers we see that $x^\rho$, as a function of $x$, has a magnitude given by $x^{\Re (\rho)}$ and its argument by $\Im(\rho)\cdot\log x$. The imaginary parts thus contribute oscillatory behavior to the explicit formulas, while the real parts say which imaginary parts dominate over others and by how much - this is some meaning behind the 'Fourier transform' description. Indeed, given the dominant term in an asymptote for $\pi$ we have roughly the primes' "expected positions" (we are taking some license in referring to positioning when we are actually speaking of distribution in the limit), and the outside terms will speak to how much $\pi$ *deviates* from the expected dominant term as we take $x$ higher and higher in value. If one of the real parts differed from the others, it would privilege some deviation over others, changing our view of the regularity in the primes' distribution.

Eventually it also became clear that more and more results in number theory - even very accessible results that belie how deep the Riemann Hypothesis really has come to be - were equivalent to or could only be proven on the assumption of RH. See for example here or here or here. I'm not sure if any truly comprehensive list of the consequences or equivalences actually exists!

Moreover it is clear now that RH is not an isolated phenomenon, and instead exists as a piece in a much bigger puzzle (at least as I see it). The $\zeta$ function is a trivial case of a Dirichlet $L$-function as well a case of a trivial case of a Dedekind $\zeta$ function, and there is respectively a Generalized Riemann Hypothesis (GRH) and Extended Riemann Hypothesis for these two more general classes of functions. There are numerous analogues to the zeta function and RH too - many of these have already gained more ground or already had the analogous RH proven!

It is now wondered what the appropriate definition of an $L$-function "should" be, that is, morally speaking - specifically it must have some analytic features and of course a functional equation involving a reflection, gamma function, weight, conductor etc. but the precise recipe we need to create a slick theory is not yet known. (Disclaimer: this paragraph comes from memory of reading something a long time ago that I cannot figure out how to find again to check. Derp.)

Finally, there is the spectral interpretation of the zeta zeros that has arisen. There is the Hilbert-Pólya conjecture. As the Wikipedia entry describes it,

In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts of the zeros of the Riemann zeta function corresponded to eigenvalues of an unbounded self adjoint operator.

This has spurred quantum-mechanical approaches to the Riemann Hypothesis. Moreover, we now have serious empirical evidence of a connection between the zeta zeros and random matrix theory, specifically that their pair-correlation matches that of Gaussian Unitary Ensembles (GUEs)...

The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study. The camera pans around the Common Room, passing by several Princetonians in tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theorist with sideburns. He has just been introduced to Freeman Dyson, dapper British physicist.

**Dyson**: So tell me, Montgomery, what have you been up to?

**Montgomery**: Well, lately I've been looking into the distribution of the zeros of the Riemann zeta function.

**Dyson**: Yes? And?

**Montgomery**: It seems the two-point correlations go as... (turning to write on a nearby blackboard): $$1-\left(\frac{\sin\pi x}{\pi x}\right)^2$$

**Dyson**: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix?

(Source: *The Spectrum of Riemannium*.)

If so inclined one can see the empirical evidence in pretty pictures e.g. here.