The main difference between Hamilton and Lagrange/Newton mechanics is, that it happens directly on the phase space, i.e. any point on your manifold already fully determines the state of your system. Intuitively, you realize this by specifying position and momentum coordinates. On a mathematical level, the world we see is some smooth manifold (a priori not even necessarily Riemann), this is the place where you specify your position coordinates. To specify momentum coordinates, you have to consider points from the cotangent space, therefore the structure where you can fully specify your state is the full cotangent bundle of your original manifold.
However, the cotangent bundle may be regarded as manifold on its own right (with twice the dimension of the original manifold). Furthermore, you can construct a canonical(!) symplectic structure quite easily by using the derivative of the natural projection. The reason, why it makes sense to put the "original manifold" aside and operate only on symplectic manifolds resp. cotangent spaces, is Darboux' Theorem, which basically says that every symplectic manifold is actually locally equivalent to the cotangent space of some manifold. Polemically simplified, one could say "The cotangent bundle is the symplectic manifold".
Lets now take a closer look at the symplectic structure: Locally, it looks like
$$
\omega=d\vec{q}\wedge d \vec{p}=dq_i\wedge dp^i
$$Using this, you can construct a volume form (up to some constant) as $\omega^n$ where $2n$ is the dimension of your manifold. Especially, you see that this volume is even oriented - locally, you can actually imagine some region as a region in euclidean space. The famous Liouville theorem states, that the Hamiltonian phase flux (i.e. the one-parametric group of the corresponding autonomic system) leaves the symplectic structure intact, therefore it also preserves the volume. This is an important consequence especially in statistical mechanics.
However, this is not main the point about the symplectic structure. The important thing is, that the symplectic form together with the Hamiltonian determines the trajectory:
Trajectories are defined as integral curves of the Hamiltonian vector field $X_H$, which is uniquely determined by the Hamilton function and the symplectic structure by
$$
\omega(\cdot,X_H)=dH
$$
Vividly spoken: If you throw some particle onto a symplectic manifold, it will move along the Hamiltonian vector field.
The point now about canonical transformations is, that it has the symplectic form as an integral invariant, or, mathematically speaking, it is a special symplectomorphism. Of course, this yields as consequence, that the local equations governing the trajectories are not affected. You can therefore think of a canonical transformation just as a reparametrization of some region of the symplectic manifold, just like a change from Cartesian to spherical coordinates. In this sense, the major advantage over Lagrange is, that you can simultaneously transform both $p$ and $q$, which of course leads to much easier equations in the end, as you can use canonical transformations to make all coordinates cyclic. This is the basic idea of Hamilton-Jacobi theory.
