Your equation is originally,
If you let , you have and (since X and T are functions of a single variable you should not have " ") so your equation becomes
which is what you have. Now divide both sides by XT:
the equation you want to get to.
The importance of this, by the way, is the fact that the right side now depends only of x while the left side depends only on t. In order to be equal for all x and t, each must be a constant! That allows you to separate the partial differential equation into two ordinary differential equations,
which are the same as
Of course, you have to determine possible values for and you usually do that by looking at boundary conditions on X. Notice that is an "eigenvalue" for both differential operators.