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In my ATPL books i've learned that for a variable pitch propeller, it is better to have a fine pitch for take-off (higher RPM) and a coarse pitch for cruise (lower RPM).

In this question taken from the ATPL question bank it states that a propeller will move towards the coarse pitch on take off.

Does anybody understand why the correct answer for this question is coarse pitch. Is this implying that we initially start our take-off with a fine pitch but slowly change towards a coarse pitch during the climb?

As airspeed increases, it gets easier for the prop to spin, which would cause a fixed-pitch prop to speed up.

A constant-speed prop's governor, however, will alter the prop pitch to be coarser (which makes it harder to spin) to counteract that and maintain the engine speed set by the prop lever.

An older variable-pitch (not constant-speed) prop requires you to adjust the prop lever manually as airspeed changes to get the same effect. However, such planes are rare these days.

What happens is the prop at idle will be at the maximum fine pitch limit. The prop control is set to Max RPM for take-off (it uses variation in pitch to regulate RPM) so the prop governor is set up to regulate engine rpm at redline once there is sufficient power to spin it that fast.

You add throttle to take off. The engine increases in speed, with the prop blades still in full fine pitch, until it gets to redline RPM, which is also the governor setting. When the RPM gets there, the governor now acts to keep the RPM from going any higher. It starts to move the blades to make the blade pitch more coarse to add load to the engine, and does so until the RPM stabilizes at the Max RPM governor setting. As you keep adding throttle, increasing torque, the governor will increase blade angle more, whatever is required to keep the RPM at the governor setting. Also, as you start to move, the forward speed starts to unload the propeller by reducing angle of attack, and the blades will coarsen further as required.

The governor doesn't know or care what the blade angle is; it's just a flyweight device like on an old steam engine and all it knows is "speed too high, do this, speed too low, do that". All the prop control does is apply a preload to a spring that determines the speed at which the flyweights will do their thing.

So yes the propeller will move toward coarse pitch because it's already fully fine and the governor has to move it more coarse to limit RPM in that situation.

If you shove the throttle in quickly, causing the RPM to rise very quickly, there may be a slight lag in the governor's response and the prop will overspeed momentarily a little bit (enough to create a noticeable EAAAOOOOWWW sound as RPM passes redline a bit and then slows down as it recovers to the governor setting).

The question is poorly worded. It should be referring to "constant speed" governed propellers, not variable pitch, which is manually operated and quite rare (way too much work). And the statement "best angle of attack" really means "angle of attack required to load the engine such that RPM is stablized at the Max RPM setting" (normally redline RPM for takeoff). And this will always be more coarse from full fine pitch.