The 4th dimension is somewhat intuitive if you think about it like a 3D space + a time dimension.
And then you have rotations, rotating space into time and vice versa is one of the most confusing things you can do with dimensions.
I find a more effective attempt at imagining 4D space to consider what it would be like to only see in 1D projection of 2D space.
Imagine standing on a piece of paper with other pieces of paper sitting on top that create a maze - all you can see as you turn and move about in 2D space are where walls and hallways are. You have a concept of depth (in 2D). You can move about the maze.
Now imagine the sensation of that flat world expanding into 3D space and your senses expanding with it. Now it's a 3D maze and you can fly- able to navigate the 3D space.
That change you just imagined, try imagining that same expansion into a fourth spatial dimension and it being projected into a 3D space you can perceive.
Instead of being able to tell the 3D shape of something by observing shading and how light interacts with it, you'd be able to perceive the 4D shape of something, even though you can only see a 3D projection at any given time.
Imagine standing on one side of a room at t0 and walking across the room, reaching the other side at t1. At each instant in time, your body has a 3D position in the room. So the 4D shape of your body between t0 and t1 is kind of like the path your body took through the room. You can create a 3D slice at any point between t0 and t1 and get exactly where your body was at that moment in time.
That doesn't help me understand 3D space. It helps me understand 2D changing over time.
Think of all the implications that "depth" - that extra spacial dimension - has on the world. How dramatically it changes things, and how differently things work with it.
If you treat time as a dimension then you now understand 3D.
There is a book called Flatland that takes place in a 2D world occupied by 2D shapes. The shapes can only perceive two dimensions. One shape encounters a sphere, but to the shape, the sphere looks like a changing circle. It makes you realize that it's probably impossible to conceptualize a sphere if your only frame of reference is two dimensions. Similarly, in your maze example, you're placing one piece of paper over another one, meaning the paper already exists in 3 spatial dimensions. How can I place one 4D object "over" another one?