Low pitch advantage
A lower pitch allows the blade to spin faster ($\omega$) for the same torque ($T$). The relationship between power ($P$) and speed and torque is $P = T \omega$, so this means that for the same $T$ we get more power out of the prop when it spins at a faster $\omega$. More power means a higher climb rate.
At low airspeed, significant sections of a high-pitch propeller will be stalled. This translates into lost thrust, and explains why a plane with a climb prop has a shorter ground roll than the equivalent with a cruise prop.
High pitch advantage
Lower tip speed
Efficiency goes down at high tip speeds, and basically falls off a cliff when the tip speeds reach the speed of sound. So a higher pitch allows for creating the same thrust at a lower tip speed, and thus gives higher efficiency.
A secondary effect is that a prop which is spinning more slowly typically leads to a more efficient engine, and reduces wear. So while this isn't directly related to your question about the prop being more efficient, it is a very important effect in the overall system efficiency.
Do propeller coefficients go to zero at the same value of advance ratio? gives a good explanation of propeller behavior for different pitch.