A passage of music that is described as allargando should be gradually slowing down. It should also begin to take on a more dignified or solemn vibe.
Allargando literally means "becoming wider or broader" in Italian, from the root largo, meaning "wide" or "broad." But what does becoming wide have to do with musical sound? Mainly, it means slowing down, as if you're gradually "stretching out" the melody like an elastic band. But it should still sound like something that flows and has power — not like a machine running out of fuel, but like a river whose current gets slower as the river broadens out. The result is a more serious or dignified sound.