Truly to visualize it is difficult, but there are a number of points that can help you think about it.
It's always good to start thinking in a space you can visualize and work from there. If you live on a curved surface, you can detect the curvature just by measuring distances and angles. For instance, on the Earth, if you measure a circle with a radius of ten thousand km, you will find that the circumference is less than $2\pi$ times the radius (where you measure both the circumference and the radius along the surface). That tells you that you are not on a flat surface. This means that, mathematically speaking, if the way you measure distances and angles changes in the right ways, or fails to follow the rules of Euclidean geometry, then you can use the mathematics for handling curvature, and the space (or space-time) is considered to be curved, even if there is no higher-dimensional space for it to be curved into.
This in turn means that, when you hear time is slower close to a black hole, or that distances have to be measured differently, that is the curvature. You can visualize it a little bit by taking a 2D slice and bending it in 3-space so the geometry works. That helps you see why it's called "curvature". But the essence of it is in the changing of time and distance.
Then you have to ask how the changing of time and distance leads to gravity. One point that it took me some time to understand is that, in space-time, you are always moving, mostly into the future. Motion in space consists of tilting your headlong path into the future so that it slopes in a spaceward direction as well.
In free fall, this path into the future will be as straight as it is possible to be. Depending on your convention for defining space-time distance, this means it should be the shortest space-time distance between two points (in space-time). This, finally, means that heading into the future without moving closer to a gravity source won't be the shortest path in space-time. (Actually, the most natural way to measure space-time distance is the amount of time experienced as you go from one event to another, and in that case the closest thing to a straight line is the one that maximizes the distance. This is surprising, but it comes from the odd way you have to measure distance in space-time in order to match the physics we observe.)