Your question was specifically about speed affecting climb performance so this is how I explain it in uncomplicated terms to non-flyers; they seem to be happy:-
In a still air hover, nil airspeed, the rotor disc is working very inefficiently as it sucks air down from above; this is caused by induced drag [same idea as needing coarse pitch at speed with a VP prop].
As the disc flies forward faster - just like a frisbee - it becomes markedly more efficient. This process starts with a "step" of reduced drag at a point known as translational lift and then continues progressively. This also applies hovering into a reasonable wind.
As the airspeed builds, however, drag on the airframe itself increases and the power required just to go faster increases.
The net result is very high power required to hover with a low power requirement in the middle speed range and high power again at the top end of the range. The typical "efficient" speed range for an aircraft capable of 120kt will be in the order of 20kt - 80kt and the most efficient will be 55-70kt depending on type.
Clearly if you want to climb at the best rate then you accelerate to it at low level and then climb, as a power loss at 50-150ft or thereabouts with no speed would result in no time to set up a stable autorotation before you hit the ground.
It's rather a long subject but that's it in a nutshell.