This problem is easiest to solve if we know a few simple number properties rules:

even * even = even
even * odd = even
odd * odd = odd

This means that for ab to be even, a and b can both be even OR one can be even and one can be odd. For ab to be odd, a and b must both be odd. For our statement to be insufficient, we must be able to make ab BOTH even and odd.

Statement 1 tells us that (a+1)b = even. We could use number properties again, but instead let's try plugging in a couple options to try to make ab BOTH even and odd.

a = 1 and b = 1 (ab = odd)
(1 + 1)1 = 2 EVEN

a = 1 and b = 2 (ab = even)
(1 + 1)2 = 4 EVEN

Both options make statement 1 true, but they make ab both even and odd. So we don't know whether ab is even. INSUFFICIENT.

Note: you can solve this problem by plugging in numbers without knowing number properties rules, but it makes it much easier to pick good numbers (and solve the problem faster) when you know these rules. Even/odd problems come up pretty frequently on the GMAT, so I highly recommend committing these rules to memory!