Each station allows you to claim a single route.
That route may be used to fulfill more than one destination ticket, but every ticket must use the same route.
From the rules,
A Train Station allows its owner to use one, and only one, of the routes belonging to another player, into (or out of) that city to help him connect the cities on his Destination Tickets.
If a player uses the same Station to help connect cities on several different Tickets, he must use the same route into the city with the Station for all of those Tickets. The Train Station owner does not need to decide which route he will use until the end of the game.
For example, say you had the following three destination tickets:
- Frankfurt – Smolensk (Complete save for Frankfurt – Berlin)
- Amsterdam – Wilno (Complete save for Frankfurt – Berlin)
- Essen – Kyiv (Complete save for Essen – Berlin)
With a station at Berlin, you could complete either the first two destination tickets or the third, but not all three (unless you also had a station at Essen or Frankfurt).
The design of the station minis helps calculate the final points. At the end of the game, move each of your stations from its city onto the route it is claiming (on top of the cars). Then that one route counts as yours for the purposes of completing tickets. You don't get to move the station again, so you can't claim multiple routes. (Credit to @Max for this tip.)