[bkfunk]

Power is important if you want to distinguish between noise and true signals. In other words, if you care about P(real effect | significant).

From good ol' Bayes we get:

P(real|sig) = P(sig|real) x P(real) / P(sig)

P(sig|real) is the power; so if you have more power, all other things being equal (a bit of a weaselly caveat), the likelihood that your stat sig result is real is higher.