How can empty space, which has no mass and is therefore not matter, curve? And how can it have an affect on the path of objects? In other words, how can empty space – which is nothing – actually do something (like curve) and how can nothing affect something?
That is an excellent question, and one that is difficult to explain. In general relativity, we talk about gravity being the effect of the curvature of spacetime. It can be difficult to imagine what this really means. There are a number of examples which are commonly used to illustrate a curved space, for example the surface of a sphere. However, when thinking about the surface of a sphere, you normally have the sphere underneath to give it substance. You don't actually need this: the surface can be thought of as a separate entity that can exist whether or not there is a sphere.
When we talk about the curvature of spacetime, what we are really describing are the properties of the metric. This is the quantity that tells us the distance between points. You can define the distance between points whether or not there is anything in between. Try to imagine two objects in a vacuum, even though there is nothing filling the gap between them, the gap could be 5 metres or 5 light-years, and that could be definitely measured. The metric exists whether or not the space is empty. In general relativity we treat the metric as a field, a physical quantity that varies with position. This isn't matter, but it is a something that does exist in a vacuum, and can be thought of as a representation of the gravitational field. You should think of spacetime (the structure of which is given by the metric), rather than a vacuum, as being curved.
Finally, how does the curvature effect matter? Matter always wants to travel in a straight line: what we mean by straight though, isn't what you might usually think of. In this case, we mean the shortest line that joins two points. For a flat space that is straight as you'd normally imagine, but try it on the surface of a sphere and you will get something that looks curved. We call these shortest paths geodesics. It takes a force to push an object off its geodesics, so when travelling unaffected through a vacuum, an object will continue happily along its geodesic. That this might look curved is just an effect of the metric, but the object would have no way of knowing without interacting with something.
I hope that goes some way towards helping you understand. Unfortunately this is a difficult subject. You can test that action at a distance works just by dropping something: it'll travel towards the Earth, even though there is nothing connecting them.