Outside of your turn you can switch from walking to flying.
So what if you suddenly want to use a different speed; how exactly does this shift from one speed to another work?
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move.
Imagine the following scenario: A creature, who has a fly speed, but is currently standing on the ground, gets launched into the air, or has the ground falls out from under them.
This is actually a really good question. Can you switch to a state of flying outside of your turn? It is wrong to assume that not being able to switch to a new speed means you can't change from walking to a state of flying or swimming. What you quoted is under the subheading "using different speeds" found under a larger subheading "breaking up your move," therefore it is specifically referring to your movement within your own turn "If the result is 0 or less, you can't use the new speed during the current move." It has nothing to do with what your character can do outside their turn. Frankly, it doesn't even say you can't switch from walking to flying, it says you cannot switch to the new speed; put plainly, you cannot switch from walking speed to flying speed. Speed is a rate of displacement, flying speed is the rate at which you can displace your own position using flight. You then cannot displace yourself using the new mode of movement, because you cannot use that speed.
Then, is there anything that refers to your movement out of turn? Under movement and position it does say:
In combat, characters and monsters are in constant motion, often using movement and position to gain the upper hand.
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
The first part of the quote refers to how creatures are moving in combat, they are in constant motion, they don't stall in their movement to allow the enemy a turn. They can change how they are moving however they want to gain the upper hand. The second part cements what I said earlier, your speed is used in reference to movement, or displacement you can achieve on your turn.
You can 'move' however you want outside your turn, you just can't achieve a displacement of position which is what your speed is.