Truly to visualize it is difficult, but there are a number of points that can help you think about it.
- 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 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, but the essence of it is in the changing of time and distance.