Learn Danish: Vi endte med at blive hjemme. - We ended up staying at home.

In Danish, the phrase for "to end up doing something" is ende med at (literally "end with to"). You need the preposition med before the infinitive marker at to make this construction work.

English uses the -ing form ("staying"). Why does Danish use at blive?

Danish doesn't use an -ing form (the present participle) in this context. After prepositions like med, Danish requires the infinitive form (at + verb). So, "ended up staying" naturally translates to endte med at blive.

I thought blive meant "to become". Why is it translated as "staying"?

blive is a very versatile verb. While it often means "to become", it is also the standard verb for "to stay" or "to remain" when referring to a location, as in blive hjemme (stay at home).

Could I say hjem instead of hjemme in this sentence?

No, because hjemme indicates a stationary location (being at home), whereas hjem implies motion towards a destination (going home). Since staying describes a lack of movement, you must use the location form hjemme.