Here's how I understand it. Classical mechanics is done on a phase space M. If we are trying to describe a mechanical system with n particles, the phase space will be 6*n*-dimensional: 3*n* dimensions to describe the coordinates of particles, and 3*n* dimensions to describe the momenta. The most important property of all this is that given a Hamiltonian function $H:M\to \mathbb R$, and a point of the phase space (the initial condition of the system), we get a differential equation that will predict the future behavior of the system. In other words, the function $H$ gives you a flow $\gamma:M\times \mathbb R \to M$ that maps a point p and a time t to a point $\gamma_t (p)$ which is the state of the system if it started at p after time t passes.
Now, if we do classical mechanics, we are very interested in changes of coordinates of our phase space. In other words, we want to describe M in a coordinate-free fashion (so that, for a particular problem, we can pick whatever coordinates are most convenient at the moment). Now, if $M$ is an abstract manifold, and $H:M\to\mathbb R$ is a function, you cannot write down the differential equation you want. In a sense, the problem is that you don't know which directions are "coordinates" and which are "momenta", and the distinction matters.
However, if you have a symplectic form $\omega$ on M, then every Hamiltonian will indeed give you the differential equation and a flow $M\times\mathbb R \to M$, and moreover the symplectic form is precisely the necessary and sufficient additional structure.
I must say, that although this may be related to why this subject was invented a hundred years ago, this seems to have little to do to why people are studying it now. It seems that the main reason for current work is, first of all, that new tools appeared which can solve problems in this subject that couldn't be solved before, and secondly that there is a very non-intuitive, but very powerful, connection that allows people to understand 3- and 4-manifolds using symplectic tools.
EDIT: I just realized that I'm not quite happy with the above. The issue is that the phase spaces that come up in classical mechanics are always a very specific kind of symplectic manifold: namely, the contangent bundlde of some base space. In fact, in physics it is usually very clear which directions are "coordinates of particles" and which are "momenta": the base space is precisely the space of "coordinates", and momenta naturally correspond to covectors. Moreover, the changes of coordinates we'd be interested in are always just changes of coordinates of the base space (which of course induce a change of coordinates of the cotangent bundle).
So, a symplectic manifold is an attempts to generalize the above to spaces more general than the cotangent bundle, or to changes of coordinates that mess up the cotangent-bundle structure. I have no idea how to motivate this.
UPDATE: I just sat in a talk by Sam Lisi where he gave one good reason to study symplectic manifolds other than the cotangent bundle. Namely, suppose you are studying the physical system of just two particles on a plane. Then, their positions can be described as a point in $P = \mathbb R^2 \times \mathbb R^2$, and the phase space is the cotangent bundle $M=T^* P$.
Notice, however, that this problem has a lot of symmetry. We can translate both points, rotate them, look at them from a moving frame, etc., all without changing the problem. So, it is natural to want to study not the space M itself, but to quotient it out by the action of some Lie group G.
Apparently, $M/G$ (or something closely related; see Ben's comment below) will still be a symplectic manifold, but it is not usually the cotangent bundle of anything. The difference is significant: in particular, the canonical one-form on $M=T^*P$ will not necessarily descend to $M/G$.