A general affine coordinate change in 3d is Ax + b, where A is a 3-by-3 matrix and b is a 3d vector. If you add a fourth coordinate that you set to 1, you can write it as a 4-by-4 matrix. That's homogeneous coordinates.

And what this means geometrically is that the translation becomes a shear in the 4d space, which is a linear transform. It’s easier to visualize if you think in 2D; the matrix

  (1 b) 
  (0 1)