In layman's terms.
1-sphere is just a circle. 2-sphere is the regular run-of-the mill sphere.
In both of those cases you're concerned with just the boundary of respectively a 2D disc and a 3D ball.
For a disc to have a meaning, you need 2D space - you need to be able to walk forward and back, left and right to reach every place on the disc. But the boundary doesn't need to be embedded in 2D space to have a meaning. All you need to travel the entirety of a circle, is go forward or back - i.e. the space you're in has just one dimension. You can, if you like, imagine the circle embedded in 2D space, which makes it easy to intuitively see the meaning of the radius of the circle. But it's not necessary. The radius can exist just as an abstract concept that tells you how long you have to go in either direction before you get back to where you started. Which is to say the embedding is optional.
Same goes for the 2-sphere, which is the boundary of a ball in 3D space. You need 3D space to reach every point of the ball - you need to be able to go forward and back, left and right, and up and down.
But the boundary doesn't need the 3D space to have a meaning. All you need to reach every spot on the surface of a ball is to be able to go left or right, up or down. I.e. the surface has two dimensions.
Again, there's the concept of the radius, which is easy to picture if we imagine the sphere as being embedded in 3D space. But, again, you don't explicitly need 3D space to fully define the sphere - the radius can very well exist just as a concept that tells you how much straight-line walking on the surface you have to do before coming back to where you started.
And again this is the same for the 3-sphere. You could try to imagine embedding it in some 4D space that requires four kinds of independent directions to travel to every point inside the 4-dimensional ball of which the 3-sphere is a boundary. But you don't need to. You can have just the 3D space of the boundary, which curves back upon itself in a manner defined by the radius of the 3-sphere. The fourth dimension needn't have any physical meaning to have a physically meaningful 3-sphere.
Now, you can't have n-spheres be flat spaces. For any n=>2 sphere to be a sphere, parallel lines can not stay parallel. For n<2 spheres the concept of flatness makes little sense, as you can't draw parallel lines on a circle or a line segment.
