Anigozanthos, aka kangaroo paws. They come in a huge range of colors and sizes.

Does it need any kind of care? Or do we just leave them alone?

Doesn't need much - just decent sun, some heat and minimal watering. I grow these as a summer annual in my area as they generally won't winter over and all I do is cut off the finished bloom stalks.

How far down should the the spent bloom stalks be cut? (sorry, we have no idea what we're doing, but we're trying to learn!)

Nice plants ... when you water them, water then thoroughly. They don't need much besides dirt, sun and water. No pampering.

And cut the dead bloom stalks down to the level of the leaves.