Learn Hardcore Russian: Я не успел на поезд, потому что долго ждал такси у вокзала. - I didn’t catch the train because I waited for a taxi at the station for a long time.

Успеть is most commonly used as a perfective verb meaning to manage (to do something in time) / to make it in time—a single completed outcome.
So Я не успел на поезд = I didn’t make it to the train (in time).
Успевать (imperfective) is used for repeated/ongoing ability to be in time, habits, or general statements, e.g. Я обычно успеваю = I usually make it (in time).

What does успеть на поезд literally mean? Why на?

In this context, успеть на + Accusative is an idiomatic pattern meaning to catch (a train/plane/bus).
So на поезд (Accusative: поезд) indicates the transport you’re trying to catch:

успеть на поезд = catch the train
успеть на автобус = catch the bus
It’s not “onto the train” literally; it’s a set way of expressing catching it.

Can I also say Я не успел к поезду?