@neatlobster Great question! Think about the grid box corresponding to a number N. The box has a purple number (N mod 5) and a yellow number (N mod 6).
We want the yellow number to be even. Write N as 6a + b, where b is N mod 6. For instance, if N is 19, we'd let a = 3 and b = 1. But because any multiple of 6 is even, this implies that 6a is always even. And we want the yellow number, which is b, to be even. So, 6a + b = N must be even since both 6a and b are even. (Basically, the whole point of the bolded sentence is to show that N is even)
From there, we want the purple number (N mod 5) to be even. BUT we already established that N is even! Therefore, N = 5*(even number) + (even number). (If N = 5*(odd number) + (even number), N would be odd, which can't be true)
Hope this explanation helped!