Learn Turkish: Birinci perondaki tren kalkmak üzere. - The train at the first platform is about to depart.

If you just used the locative case and said peronda, it would mean 'at the platform' as a general location (like 'A train is departing at the platform'). Adding -ki turns that location into an adjective attached to a specific noun. Therefore, perondaki tren means 'the train that is at the platform'.

I thought kalkmak meant 'to get up'. Why is it used for a train here?

You're right that kalkmak means 'to get up' or 'to stand up' when talking about people. However, it is also the standard verb used for public transport—such as trains, buses, ferries, and planes—to mean 'to depart' or 'to take off'.

How does the grammar work to create the 'about to' meaning in kalkmak üzere?

To say you are 'about to' do something, you use the full dictionary form of the verb (ending in -mak or -mek) followed by the word üzere. This structure (-mak üzere) always means that someone or something is on the verge of doing that action right now.

Since the English translation says 'is about to depart,' where is the 'is' in the Turkish sentence?

The 'is' is implied at the very end of the sentence. In Turkish, the third-person 'to be' suffix (-dir) is frequently dropped in everyday speech and writing. If this were a formal train station announcement, you would likely hear the official, full version: kalkmak üzeredir.