Think of fine (less) pitch as 1st gear in a car; coarse (more) pitch as 5th gear. For take off you want fine pitch just as you'd want 1st gear for starting off in your car, or even for climbing a very steep hill. Once you're actually underway and picking up speed you want the prop to "bite" more air to keep increaing your speed and to hold that speed, just like the upper gears of your car.
A constant speed prop (Spit Ia & IIa, Hurri Rotol) does not act like a manual shift car's gearbox, nor like a car's automatic transmission either. As its name implies, a contant speed prop has a speed governor mechanism that works to keep the propeller turning at a constant speed which translates to the engine turning at a constant speed. This speed governor adjusts the pitch of the prop to do that. If you open the throttle (increase engine boost), the propellor pitch will coarsen slightly to maintain the engine speed you have set with your Prop Pitch Control Lever.
Here's an example: say you are flying along in level flight at 200 mph with your throttle 3/4 open (giving you engine boost of, say, 1.0 lb) and you've set your rpms to 2500 using your Prop Pitch Control Lever. Suddenly you push your throttle fully forward (wide open). What happens?
Your engine boost will suddenly increase to 6.0 lbs. The prop's speed governor will try to keep your engine speed at a contant 2500 rpms, so it will quickly coarsen your prop pitch, putting a heavier load on the engine in so doing. The rpms stay at 2500, but now -- because the coarser prop pitch is taking a bigger bite of air --- your speed suddenly increases. You'll also find yourself climbing unless you adjust your elevator pitch, and you'll be rotating to the left because of greater engine torque. Once you've gotten yourself trimmed back to level flight you'll notice you're travelling faster. Much faster. Enough to catch a bomber!
