Learn Hardcore German: Wir mussten am Bahnsteig umsteigen, aber der Zug fuhr in Eile ab. - We had to change trains on the platform, but the train departed in a hurry.

umsteigen literally means “to change” (trains, buses, etc.)—that is, to get off one vehicle and onto another.
Use it whenever your journey requires a transfer:

Ich steige in München um. (I change trains in Munich.)
Wir müssen am Flughafen umsteigen. (We have to change at the airport.)

Why is umsteigen written as one word, and why isn’t the um- prefix separated here?

German separable-prefix verbs (like umsteigen, abfahren, ankommen) are written as a single word in the infinitive and in non-finite forms. The prefix only detaches in finite verb forms:

Present: ich steige um
Präteritum: ich stieg um
But in the infinitive (here after a modal verb) it stays together: umsteigen.

Why is the main verb umsteigen at the end of the clause after mussten?

In German, when you use a modal verb (müssen) in a main clause, the finite form of the modal takes the second position (here mussten), and the main verb goes into its infinitive form at the very end:
… mussten … umsteigen.

Why do we say am Bahnsteig? What does am stand for, and what case is Bahnsteig in?

am is a contraction of an dem, where an is the correct preposition for being “at” a platform. Bahnsteig is masculine (der Bahnsteig), so in the dative case it becomes dem Bahnsteig → am Bahnsteig.

What exactly is a Bahnsteig, and how is it different from a Bahnhof?

Bahnhof = station (the entire train station building or complex).
Bahnsteig = platform (the raised area beside the tracks where you board or leave a train).

Why is there a comma before aber? I thought English coordinate conjunctions didn’t always need one.

In German, when aber (as well as denn, sondern, doch) connects two independent main clauses, you must place a comma before it:
Wir mussten umsteigen[,] aber der Zug fuhr in Eile ab.

Why does the sentence use simple past fuhr ab instead of the perfect tense ist abgefahren, and how does the separable prefix ab behave here?

German often uses the Präteritum (simple past) in written or narrative contexts.
For the separable verb abfahren, the past stem is fuhr, and the prefix ab goes to the end:
Der Zug fuhr … ab
In the perfect you’d say Der Zug ist abgefahren (prefix attached in the past participle).

What does in Eile mean, and why isn’t there an article like in der Eile?

in Eile is an idiomatic expression meaning “in a hurry” or “in haste.”
Some fixed expressions with Eile drop the article. You could say in der Eile, but that sounds more formal or poetic; everyday German uses in Eile.

Can I replace in Eile with an adverb like eilig, as in Der Zug fuhr eilig ab, and is there any difference?

Yes. eilig is an adjective used adverbially (“hurriedly”), so Der Zug fuhr eilig ab is grammatically correct.

in Eile focuses on the state of being in a hurry.
eilig describes the manner (“the train departed hurriedly”).
Both are fine; in Eile is just a very common idiom.