You don't actually need to know too much, just the basics. A Matrix can rotate and scale a vector. It does this by defining 3 new axes. The axes don't have to be normalized (1 unit length) or orthogonal (at right angles to each other). Let's take some examples:
We will start with the 3x3 identity matrix, which is the matrix representation of the number 1:
1 0 0
0 1 0
0 0 1
If you look at the columns of the matrix, you get 3 vectors:
(1, 0, 0), (0, 1, 0), (0, 0, 1)
These are the unit vectors for the x, y, and z axes, respectively.
When you multiply a matrix by a vector, you are transforming the vector into the coordinates of the matrix. For example, for the identity matrix, we can do
1 0 0 3
0 1 0 x 4
0 0 1 5
gives us the first component of the vector times the first column of the matrix plus the second component of the vector times the second column of the matrix plus the third component of the vector times the third column of the matrix:
3*(1, 0, 0) + 4*(0, 1, 0) + 5*(0, 0, 1) = (3, 4, 5)
As you can see, multiplying the identity matrix by any vector simply returns the original vector.
Now let's try a simple scaling matrix that scales the x component by a factor of 3, the y component by a factor of 1/2, and the z component by a factor of 3/5. To do this, we take the identity matrix, and multiply the x column by 3, the y column by 1/2, and the z column by 3/5:
3 0 0 3
0 1/2 0 x 4
0 0 3/5 5
now when we multiply by the original vector, we get:
3*(3, 0, 0) + 4*(0, 1/2, 0) + 5*(0, 0, 3/5) = (9, 2, 3)
Now let's try a simple rotation matrix that rotates the vector around the z axis by 90 degrees. To do this, the old x component will be the new y component, and the old y component will be the new -x component:
0 -1 0 3
1 0 0 x 4
0 0 1 5
now when we multiply by the original vector, we get:
3*(0, 1, 0) + 4*(-1, 0, 0) + 5*(0, 0, 1) = (-4, 3, 5)
In general, you can combine the scaling and rotating effect, and you can see matricies like this:
0 -1 3 3
1 0 0 x 4
2 4 1 5
3*(0, 1, 2) + 4*(-1, 0, 4) + 5*(3, 0, 1) = (11, 3, 27)
There's a lot of stuff I'm leaving out (like translation), but this is a good crash course that should let you understand most of the things you need to do.