KV is "RPM per volt", so higher KV will give higher prop speed at WOT. For high power, the reason "Low KV/Long Prop" or "Extremely high KV/tiny prop" are popular choices is effects on blades at high speeds.
To push more air (more thrust) we need to either get a longer prop with steeper pitch, or spin it faster. However, the longer the blade, the faster the linear speed out at the tip. When the linear speed on the tip goes too high the blade stops working as designed. So we get a really long prop it'll really pull hard, but we've got to spin it slower to keep it working, hence low KV/long prop.
There is a point where we could go the other way, shortening the prop, kicking up the RPM, and still keep the tip speed down. The balance can get more thrust out of the shorter prop, but it's a bit more tricky and has harder limits to what you can get.
In performance, higher KV gives a more responsive thrust production -- nice for zippy flying that takes advantage of throttle management. Lower KV takes longer to change from one RPM to the next -- something carrying FPV gear or a candy drop gets the power it needs, but the throttle will feel a little sluggish in comparison.