Physicist: Math. Math all over.
A picture of a 3D object is a “projection” of that object onto a 2D page. Projection to an artist means taking a picture or drawing a picture. To a mathematician it means keeping some dimensions and “pancaking” others.
So when you take a picture the “up/down” and “left/right” dimensions are retained, but the “forward/back” dimension is flattened. Mathematicians, being clever, have formalized this into a form that is independent of dimension. That is, you can take an object in any number of dimensions and “project out” any number of dimensions, until it’s something we can picture (3 or fewer dimensions).
Top: An object in 3 dimensions. To see it, cross your eyes by looking "through" the screen until the two images line up. Middle: By "projecting out" the z axis (toward/away) the object is collapsed into two dimensions. This is what cameras do. Bottom: By projecting out the y axis (up/down) the object is collapsed again into 1 dimension. This is akin to what a 2D camera would see, photographing from below.
We’re used to a 3D-to-2D projection (it’s what our eyeballs do). A 4D-to-2D projection, like in the picture above, would involve 2 “camera/eyeball like” projections, so it’s not as simple as “seeing” a 4D object.
As for knowing what a 4D, 5D, … shape is, we just describe its properties mathematically, and solve. It’s necessary to use math to describe things that can’t be otherwise pictured or understood directly. If we had to completely understand modern physics to use it, we’d be up shit creek. However, by describing things mathematically, and then following the calculations to their conclusions, we can get a lot farther than our puny minds might otherwise allow.
Lines, squares, cubes, hyper-cubes, hyper-hyper-cubes, etc. all follow from each other pretty naturally. The 4D picture (being 4D) should be difficult to understand.
For example, to describe a hypercube you start with a line (all shapes are lines in 1D).
To go to 2D, you’d slide the line in a new direction (the 2nd dimension) and pick up all the points the line covers. Now you’ve got a square.
To go to 3D, you’d slide the square in a new direction (the 3rd dimension) and pick up all the points the square covers. Cube!
To go to 4D, same thing: slide the cube in the new (4th) direction. The only difference between this and all the previous times is that we can no longer picture the process. However, mathematically speaking, it’s nothing special.
Answer gravy: This isn’t more of an answer, it’s just an example of how, starting from a pattern in lower dimensions, you can talk about the properties of something in higher dimensions. In this case, the number of lines, faces, etc. that a hyper-cube will have in more than 3 dimensions.
Define 
Now define 
For example, by looking at the square (picture above) you’ll notice that 
The “slide, connect, and fill in” technique can be though of like this: when you slide a point it creates a line, when you slide a line it creates a square, when you slide a square it creates a cube, etc. Also, you find that you’ll have two copies of the original shape (picture above).
So, if you want to figure out how many “square pieces” you have in a D-dimensional cube you’d take the number of squares in a D-1 dimensional cube, double it (2 copies), and then add the number of lines in a D-1 dimensional cube (from sliding).
The values of e_N(D) arranged to make the pattern clearer. You can use the pattern to accurately predict what the cube in the next dimension will be like.
It’s neither obvious nor interesting how, but with a little mathing you’ll find that 
In order to actually do meaningful math in dimensions higher than three, you have to make your peace with being entirely unable to do more than (a) draw crude two- or three-dimensional visualizations or (b) draw vague scribbly conceptual pictures.
(It’s actually kind of funny to watch topologists “prove”* things. Spaces (even spaces where “dimension” doesn’t even make sense) become boxes, or lines if you’re taking products or suspending. Embeddings become squiggles in blobs. Suspensions? Pointy spheres. Taking a quotient? Scribble in the subspace being identified. It goes on and on.)
* These things all have precise definitions. The pictures merely suggest the actual proofs, which of course are always left as an exercise.
On the hypercube, are the angles also 90 degrees, like in a regular cube and in a square?
Assuming yes: so there are 4 sides which all are perpendicular to each other?
Assuming no: what are the angles between the sides? can they be measured?
Yes.
In an N-dimensional cube each corner is the intersection of N mutually perpendicular sides.
The fourth dimension is not at all hard to conceptualize from the drawing. If one is used to seeing three dimensional objects move in any of the three dimensions, and can remember that at time t the 3D object was here, and now at time t+1, the 3D object is now THERE, you basically have it.
Intuitively conceptualizing the fifth dimension from the first four is the tricky part because we have no intuitive frame of reference. I imagine it would be akin to watching a 3D cube move in one direction, while its 6 shadows on the sides of a larger box containing it from the six coordinate light sources also move, and you would see the lines of the shadow persisting from each opaque line itself to the walls of the cube containing the 3D cube.
Of course, this is mere intuition.