The SVR performs linear regression in a higher (infinite) dimensional space. A simple way to think of it is as if each data point in your training set represents it's own dimension. When you evaluate your kernel between a test point and a point in your training set, the resulting value gives you the coordinate of your test point in that dimension. The vector we get when we evaluate the test point for all points in the training set, (\vec{k}), is the representation of the test point in the higher dimensional space. The form of the kernel tells you about the geometry of that higher dimensional space.
Once you have that vector, you use it to perform a linear regression. You can tell it is a linear regression because of the form of the estimator, it's an inner product! Thus the intuition for the machine is rather simple, though the procedure and parameters may be difficult to interpret.
Because you are able to generate training points, you know what the "right" answer is. However, it may be very expensive to compute that answer for every new point you need. The SVR, and in particular Gaussian processes, are very good at providing a cheap surrogate to an expensive call. If the function you are trying to compute is smooth, and you expect to have to call it over and over again, you may be able to gain a significant savings by pre-computing a training set and then using a SVR machine to interpolate the results.