Why does matrix multiplication seem so unintuitive and weird?!

You're right, when you first learn about matrix multiplication it definitely doesn't feel like the easiest way of defining a form of multiplication.  This is particularly true if you've just learned about adding matrices and multiplying them by scalars, which do feel intuitive - surely it would be simpler to define multiplication by:

| a c | | w y |  =  | aw  cy |
| b d | | x z |       | bx  dz |

as opposed to the the normal way.

On some level it doesn't really matter!  Mathematics can be whatever we think of, so if you want matrix multiplication to be like this, the go ahead!  You might find some interesting maths if you do this.  The fact that it isn't defined like this, however, is probably a good indication that there's a reason for it being defined the normal way.  Mathematicians are smart people, after all!

The reason for matrix multiplication is because of what a matrix signifies.  To a computer scientist, a matrix is just a 2D array of numbers.  To a mathematician however, they signify something much greater: matrices represent a type of transformation, known as linear transformations because of how they behave.  When we perform matrix multiplication, what we are doing from a visual perspective is to ask what happens when one transformation is followed by another.  The nature of linear transformations mean that we can write the combination of these matrices as another matrix, and to work out what this overall transformation looks like we follow the rules of matrix multiplication.