Translation means moving a point from one position to another. Assume the point's original homogeneous coordinates are (x,y,z,1), and it is to be translated dx, dy and dz respectively. This means that the new position will be (x+dx, y+dy, z+dz, 1). In this way, the point is translated by adding something to each coordinate. It is more desireable however to find a way to multiply a point with a square matrix to get the new position rather than add to its coordinates. Since all the other transformations (like rotation and scaling) can be done with a matrix multiplication, we want to bring translation into line with them. Why? Because if we do this, we can perform a series of transformations on a point by multiplying all the transformations together before multiplying the point itself.
The example below shows what the square translation matrix looks like that multiplies the row matrix (the original coordinates of the point) to move a point. You can see how homogeneous coordinates make this multiplication possible. Without creating that 1 as the fourth element of the point's coordinate matrix, there would be no way to multiply this row matrix to get a translation. So, in order to move a point any amount in any of the three axis directions, you have to replace the dx, dy and dz with the change information and multiply the row matrix of the original point by that translation matrix. if there is no movement in a given direction, for example if the point moves in the x and y direction but does not move in the z direction, then the dz element in the translation matrix would be zero.