wlog, let y>=x

For (x+y)>4, (x+y)! ends in 0

So if (x+y)>4, the RHS ends in zero, therefore and one of x! or y! must end in 4, the other must be odd to get a LHS with an even number times a number ending in 5.

The only odd factorial is 1 which is 0! or 1!

The only factorial ending in 4 is 24 which is 6!

(1!+1)(4!+1) = 50 which does end in zero but is not 5!

There is no other way for the LHS to end in zero.

Thus x+y <= 4 so there are only 15 (x,y) pairs that pass this test (only 9 if y>=x)

This is small enough to check all manually and determine that the only solution is:

But a little further analysis narrows it down without testing all 9:

The factorials of 0,1,2,3,4 end in 1,1,2,6,4

Adding one, each factor on the LHS can end in 2,2,3,7,5

What ways can two factors chosen from {2,3,7,5} end in {1,2,6,4}?

2*2=4 

2*3=6  <-- x=1,y=2 works

2*7=14

3*7=21

A quick glance at the RHS results shows that only 6 is a factorial

And for overkill, a program:

----------------

def fact(n):

    if n == 1 or n == 0:

        return 1

    ans = 1

    for i in range(n):

        ans *= (i+1)

    return ans

for x in range(100):

    for y in range(x,100):

        if (fact(x)+1)*(fact(y)+1) == fact(x+y):

            print(x,y)