"Easier to find" is not quite the right thing to say. The question is this: you want to pick an extremely large number to test for primality. What kind of large numbers should you test? A "random" large number is a bad choice: such a number will have a probability of approximately $\frac{1}{2}$ of being even, a probability of approximately $\frac{1}{3}$ of being divisible by $3$, and so forth.
Mersenne numbers, on the other hand, are both extremely large and substantially more likely than "random" large numbers to be prime. Indeed, if $q$ divides a Mersenne number $2^p - 1$, then $q \equiv 1 \bmod p$ by Lagrange's theorem. This is a far smaller list of possible prime divisors, and in particular no prime less than or equal to $p$ can be a divisor.
There is also a specialized primality test, the Lucas-Lehmer test, which is specific to Mersenne numbers.