Imagine that you are in a free-falling elevator. You will feel as if you are weightless. There is no way you could tell whether you are falling in a gravitational field, or whether you are in the depths of space, far from any source of gravity. Any experiment you can do will have the same results in both cases.
Actually, there is only one way to detect the presence of gravity -- if you could look out the window and see another elevator falling beside you, you would notice the elevator slowly coming closer to you. This is because both elevators are being drawn toward the center of the earth. In the absence of a gravitational field, any two objects that are "free-falling" (have no forces acting on them) will NOT be drawn closer to each other. In a gravitational field, two objects that are free-falling (have no forces other than gravity acting on them), may be drawn closer together. This is called a tidal effect, and it is gravity's only effect.
This may seem very counter-intuitive. On the surface of the earth, we think that we feel the force of gravity. Actually, though, the force that we feel is the force of the ground preventing us from continuing on our natural, free-falling path.
Now the reason curvature enters into the picture is that the mathematical concept of curvature allows us quantify the effects of a force that is only detectable through its tidal effects. You can imagine two ants crawling in a straight line on a flat surface. If they start out going in the same direction, their paths will never cross. Now imagine they are crawling on a curved surface, like the surface of an apple. Then even if they try their best to go in a straight line, they will find that they are sometimes drawn closer to each other because of the curvature of the apple. The modern view of gravity is that all objects attempt to travel in straight lines, but that gravity bends space. Because of this, objects travelling in "straight" lines may be drawn closer to each other as a result.
Edit
You might object to the above reasoning by saying that the falling motion of the elevator is simply offsetting the gravitational effect. That is certainly one way of looking at it. However, since gravity effects all particles (unlike any other force), there is no way to tell whether the elevator is in a gravitational field or not except to look at tidal effects. It is actually conceptually (and mathematically) much simpler to take the view that all particles naturally attempt to move in straight lines, and the curvature caused by gravity can cause them to come together. This approach has led to many useful new predictions that have been experimentally confirmed. Technically, though, you could view gravity as some kind of force if you want. It's just much harder to account for all the new effects when you do it that way.
Edit 2
It can be hard to reconcile this geometrical point of view with everyday experience. As an example, consider two massive balls, at rest with respect to each other in empty space. It can be hard to understand why curvature would cause these two balls to come together, since they are not moving through space. The key thing, however, is that they ARE moving through time. In relativity, space and time are not two separate concepts. Gravity curves both space and time, so as the balls move through time, the gravitational curvature will cause them to bend into each other.
