Passenger Destination

The following information applies equally to both passengers and mail deliveries.

Preset

To enable passenger destination, you need to set it before starting new game in the Advanced preset menu. The game will remember this preset for all subsequent new games until you change it again. Please remember, you can enable/disable passenger destination ONLY when starting new game. Once the game is started, you cannot change it. If you have an old saved game from the time before passenger destination was implemented, you will have the choice to enable/disable it when opening this saved game. But the choice is permanent and can't be reversed. If you overwrite the old saved game you can never go back and play it the other way.

If you're not sure whether passenger destination is enabled or disabled in any saved game, you can find out by opening any station window or any passenger train/vehicle window:

Passenger destination disabled

Trains and buses will fully unload at each stop and also load all available passengers and mail until their capacity is reached or no more people (mail) is available. Then they will depart and the process is repeated at each stop along the route.
The station or vehicle info panel shows only the total number of people (mail) waiting / being transported:

Passenger destination enabled

Passengers and mail behave just like in real life - they want to go somewhere on the map as long as valid connection exists. They do not care about how many times they have to transfer, as long as there is the possibility to get to a place by any combination of trains and buses (mail trucks).
Of course, the trains (buses) do not unload completely at each stop, only people who are destined to that place and people who want to transfer are leaving the train. Thus, the trains will carry away less people from the stop, because they will load only the exact number to reach their capacity, as long as the people waiting there want to go someplace the train is going. It is not uncommon to see train/bus leaving not fully occupied with number of people (mail) still waiting at the stop (but wanting to go somewhere else than the departing train goes). This correct behavior and is much more closely resembling real life behavior.
You will get paid at each stop for the distance traveled for all passengers (mail) disembarking at that stop, but the transferring people (mail) do NOT count towards the city growth. They will just wait at the station for the next train/bus/truck to get them further towards their final destination. Only passengers and mail, whose destination is within the catchment area of that particular station, will leave the station and will count towards the city growth.

Warning: Do not use "Wait until fully loaded" nor "Force unload" commands for trains/vehicles with people and/or mail if you have passenger destination enabled. It does not work and it will confuse the whole system!

Each station and vehicle window shows not only the total number of people (mail) waiting, but also how many of them want to go to each particular destination. Depending on your network connections, this list could be very extensive:

Useful Hints

Playing with passenger destination enabled adds realism to the game, but it also requires you to think about your connections. Connection does not mean two stops being connected by rail or road, but rather existing train/bus service between such stops. Even if it is service where a passenger would have to transfer several times between different buses and trains.
If you connect all stops on the map into one huge network, there will be eventually people waiting on small bus stop in one corner of the map wanting to go to another tiny bus stop in the other corner of the map. Since the map could easily have more than hundred (or even hundreds) stops (train and road stops together), it could happen that from 100 people on one station, each of them would want to go somewhere else. Such transport will have very little if any efficiency.
The destination desires do not happen completely randomly, there are preferences depending on the size of the city around any given stop, train stops are preferred over bus stops, and more, but the system is not perfect and players have developed several strategies. Since Mashinky game cannot be played wrongly and there is never only one correct approach to a challenge, you should always explore different options and find what suits you the best.
One way is using the Star method - you select some large central city as a hub and connect peripheral cities to it in the shape of a star (or asterisk).
Another way is using the Cluster method - if you have several cities on the map in convenient cluster, connect them in circular (or any other shape) route.
Using any way of playing, you can get the best results if you have up to about 10 stops connected to a network. Of course it will work if you connect more or even all of them. The game will not break. But then it could happen that a train with capacity (for example) of 180 people will offload/reload less than 10 passengers at any stop, which will significantly slow your city growth.

Tooltip info

When you open and highlight any passenger (mail) train/vehicle, you can see number in blue background at the end of the train tooltip, showing the total number of passenger (mail) with destination on the train/vehicle. At the same time, you can see number in yellow background at all stops over the whole map showing how many passengers (mail) from this particular train/vehicle has destination to each of the marked stations.

Similarly, when you open any station window, all stations on the whole map, which are as destination for any of the passengers waiting at the opened station, will have number in yellow showing how many of them want to go there. It could be very useful if you want to change some connections and want to know where everywhere people want to travel from any particular city.

Lost Connections

If you delete stop from the train route list while the train is en-route, people who wanted to go there will lose their destination. They will disembark at the next station and walk away to the city.
If you delete a particular stop from all trains/vehicles, so there is no connection to this particular stop, all people at this stop will lose their destination, but they will stay at the stop. The stop info window will look like the one with passenger destination disabled.
Similarly, newly built stop will have just the total number of generated people (assuming you are building it while the game is not paused) without destinations. As soon as you set a route/connection to this stop, the destinations will be assigned.

Note: Destinations are assigned as soon as you create the route in the train/vehicle window, NOT after you send the train/vehicle en-route. If you have people waiting at the stop and wanting to go where no bus/train is going, it means that you have a train/bus with this stop in their route list parked somewhere - in the depot, on side track, in "museum", simply anywhere, but the connection still exists, so people want to go there. 
Remember always delete the route list before parking any train/vehicle. Of course, selling train/vehicle will delete the connections.