Learn Swedish: Han ändrade sig i sista minuten. - He changed his mind at the last minute.

The verb att ändra means to change a physical thing or a plan. To say someone "changed their mind," Swedish uses the reflexive phrase att ändra sig (literally to change oneself). Without sig, the sentence sounds incomplete.

To say "at the last minute," why do we use the preposition i instead of på or vid?

Swedish uses i for this set expression, which behaves exactly like the English alternative "in the last minute." You cannot translate the English "at" directly to på or vid for this specific phrase.

Why isn't there a separate definite article like den before sista minuten?

Adjectives like sista (last), nästa (next), and samma (same) are strong enough to point out a specific thing on their own. Therefore, you drop the standalone article (den or det), even though the noun still gets its definite ending (minuten).

Could I say i senaste minuten instead?

No, because senaste means the "most recent" or "latest" (with the possibility of more to come). sista means the absolute final one. Since he made his choice at the final possible moment, you must use sista.