It means assuming a solution of the form u(x,y)= A(x)B(y) where A and B are functions of x alone and y alone, respectively. Then and .
So that equation becomes . That can be written as and, dividing both sides by AB, .
Now, the left side is a function of x only and the right side is a function of y only (we have "separated" the variables) so the only way that can be equal for all x and y is if they are both equal to the same constant.
That is, we can separate into two equations: or and or .
There is no way to determine what " " is from the equation alone. Depending upon the additional (boundary or intial value) conditions, the solution might be the product of A and B for a specific or a sum of such products for many (possibly infinitely many) values of .