Learn Russian: Короче говоря, мы опоздали на поезд.

Literally, it translates to "speaking shorter." Короче is the comparative form of короткий (short), and говоря is a verbal adverb meaning "speaking." In everyday conversation, Russians often drop говоря and just say Короче to mean "In short" or "Anyway...".

Why is "missed the train" translated as опоздали на поезд instead of using a word like пропустили?

In Russian, you do not "miss" transport; instead, you are "late for" it. The verb опоздать means "to be late," so мы опоздали на поезд literally means "we were late for the train." If you used пропустить (to miss or let pass), it would sound like you intentionally watched the train go by without getting on.

Why do we need the preposition на before поезд, and what case is used?

The verb опоздать (to be late) requires the preposition на followed by the accusative case to show what you are late for. Since поезд is an inanimate masculine noun, its accusative form looks exactly like its nominative form.