If r is the LCM of p-1 and q-1, you can write :
r = α.(p-1)
r = β.(q-1)
for some integers α and β.
Let M a message, M^e is the encrypted message, and (M^e)^d is what you get when trying to decrypt with an exponent d.
(M^e)^d = M^(ed)
Since ed = 1 mod r, you can write : ed = 1 + k.r, for some integer k.
M^(ed) = M^(1 + k.r) = M.M^(k.r) = M.M^(kα.(p-1)) = M. ( M^(p-1) )^ (kα)
If M is not a multiple of p, then according to Fermat's theorem : M^(p-1) = 1 mod p.
Hence : M^(ed) = M. 1^(kα) = M mod p.
And : M^(ed) - M = 0 mod p : M^(ed) - M is a multiple of p.
Else, if M is a multiple of p, then the same result obviously stands.
Do the same with q : M^(ed) - M is a multiple of q.
Since it's a multiple of both p and q, that are coprime, then it's a multiple of their product n :
M^(ed) - M = 0 mod n
M^(ed) = M mod n.