The way I visualize n-spheres is by starting with the projection into (n-1)-dimensions, and the "inferring" nD by dimensional analogy.
For example, a 3D sphere projects into a 2D circle (well, technically, a disk -- a filled circle). Of course, a 3D sphere is certainly not "just a circle", because if you look at a picture of a sphere (which is the same as the projection of a sphere), you notice that it "bulges outward" in the center, and that half the sphere actually lies behind the circular projection image. So the shape of a 3D sphere is basically two 2D disks "sewn together" at their edges, and "inflated" so that they "bulge" into a sphere. If you drew a grid on the sphere, like this:
you'll notice that near the center of the image, the grid cells are closest to squares (or rectangles in this case). Near the edge of the circular image, they get increasingly distorted, because the curvature of the sphere is more pronounced there. They also appear more "squished", because they're increasingly seen from an angle.
What of a 4D sphere? It projects to a 3D sphere (or rather, a ball -- a filled sphere). But of course, it's not "just a 3D sphere", because a 4D viewer looking at a 4D sphere would see it "bulging outwards" at the center of the image, which, in this case, is the center of the 3D sphere. So a 4D sphere is basically the interiors of two 3D spheres, glued to each other at their boundaries, and "inflated" so that they bulge along the 4th axis into a 4D sphere.
So imagine if you will, the center of this sphere (where the lines converge) is "bulging outwards", not in the direction of the screen or any 3D direction, but in the 4th direction.
Near the center of the projection is where the shapes of the grid cells (which in this case are cuboidal blocks or pyramids) are least distorted; as you move outwards, they become more and more squished. This isn't very clear in the above image due to the distortions introduced by the convergence point, but it's clearer here:
The outermost layer of cells are the most "squished", signifying that that's where the curvature of the 4D sphere is most visible.
So there you have it, the 4D sphere.
Analogously, the 5D sphere can be visualized once you've mastered the 4D sphere: it basically projects to a 4D sphere where its center is "bulging outwards". This center is not the center you see in the images above (what you see in the images is just the surface
of the 4D sphere), but the actual 4D center of the hypersphere. So imagine that center is "bulging outwards" in the 5th direction, and another copy of the 4D sphere bulging in the opposite direction, and there you have the 5D sphere.
You can do the same for the 6D sphere, the 7D sphere, etc..