Я еле успел на поезд, потому что попал в пробку.

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Breakdown of Я еле успел на поезд, потому что попал в пробку.

• потому что: because
• еле: barely
• успеть на поезд: to catch the train

Questions & Answers about Я еле успел на поезд, потому что попал в пробку.

What does еле mean here, and how is it different from едва?

Еле means barely / just (managed to) and often has a slightly colloquial feel. In this sentence it implies you almost missed the train.
Едва is very close in meaning (barely / hardly) but can sound a bit more neutral or “bookish” in some contexts. Both can work: Я едва успел на поезд is also correct.

Why is it я еле успел, not я еле успеваю?

Because the sentence describes a completed past event: you did make it, but only just.

• успел = past, completed (“managed in time”)
• успеваю = present, habitual/ongoing (“I am успеваю-ing / I’m managing (in general), I’m on time”)

What does успел на поезд literally mean? Is there an omitted verb like “to catch”?

Russian often uses успеть to express “manage to do something in time.” With transport, успеть на + accusative means to make it in time for / catch something.
So успел на поезд = “managed to catch the train” (literally: “managed (in time) for the train”).

Why is it на поезд (accusative) and not в поезд or к поезду?