How do I make my trains use the empty platforms in my train station?

In order to make your trains use your empty platforms on this station, it may be instructive to you to read more about how signals work in depth. Signals divide your rail into what's commonly known as signal blocks.

All signals will forbid a train from entering a block if it is already occupied. Path signals (the type in front of your station) have an exception: they will allow more trains in as long as the incoming train does not share a path with any trains currently within its block. Path signals will thus take into account where trains currently are. If the path a train is trying to take within its block is currently fully unoccupied, the signal will let the train through.

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Thought experiment

Imagine what happens if there are few passengers waiting, and two trains are let into the station. Now a rush of passengers suddenly appears, fully loading both trains. Both trains will start at the same time, and crash into eachother!

We can thus say that your station is signalled as a single block, and has a single point of entry as well as exit within this block. As such, only one train will ever enter this block at the same time. Your station design cannot guarantee that trains will not share a path after they leave, so the game will only allow a single train in at a time.

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The solution should now be obvious:

Try lengthening the area occupied by one tile, and extending each rail platform by adding a single tile of rail at the back. Then build a regular signal behind each platform. Trains should now use all platforms, as the path signal will now allow trains to pass.