Let's take an example

5 / (x - 1) + 3 / (x + 2 ) = 5

The LCM is (x - 1) ( x + 2)

So

5 (x - 1) ( x + 2) / (x - 1) + 3(x - 1) ( x + 2) / (x + 2) = 5(x - 1) ( x + 2)

5(x + 2) + 3(x - 1) = 5(x - 1) ( x + 2)

So...multiplying by the LCM "cancels out" the denominators