From page 214 of the DMG under the description for the boots:
While you wear these boots you have Flying Speed. Flying speed does not make you fly it allows you to fly when you want to, you have to willingly decide to fly.
From the Player's Handbook page 196 (emphasis mine):
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speed during your move.
Also it states:
...you can use the boots to fly up to 4 hours...
So if you use the boots by willingly deciding to move while flying, during your move then you are flying.
But the falling Optional rule from Xanathar's Guide to Everything (pg78) book needs to be taken into account here.
... you immediately fall 500ft...
Immediately means you are not allowed to do any action other than a reaction since the fall is interrupting your current turn and the only activity you can do out of your turn is a reaction, not a movement, not an action, not even a free action. Using the boots is definitely not a reaction, else it would be stated RAW in the magic item that it can be used as a reaction as an exception (like the way featherfall is written).
So, if during your turn you decide to walk (to avoid tapping into the 4 hour total duration) and move unto a trapdoor that opens under your feet; you immediately fall 500ft, then once you are 500ft lower, you can finish your move, and switch from walking to flying if you have not hit the ground yet. Otherwise you take falling damage, are prone, and can finish your movement and any action or bonus action remaining.
The same effect would happen if during someone else's turn your are pushed off a cliff. Since it is not your turn, you cannot move, therefore cannot switch from walking to flying, you fall 500ft immediately.
But if you were already flying (thus activated the 1 minute minimum flying time) then you are ok, because you are actually in flying mode and are not even touching the ground.
This is per RAW rules, not RAI. Some DM might want to allow some 'reaction' to start flying with the boots and I guess within RAI games that could be ok, but in RAW games I don't see how that is possible.