How I can incorporate this piece into our rustic nursery?

Do you have a better picture?

It's already cute. I'd just paint it to go with the nursery , maybe distressed. maybe add stickers to decorate it and hang cutie baby outfits on it.

I think this will be cute for hanging dress up clothes later on, but for now, you can hang blankets, hooded towels, clothes, etc. You could paint it to go with your room colors, but if you want to keep it to where you can move it around later and it's more neutral, you can just add some easy on/off decals etc to go with the room now!

When I want a rustic, distressed look I always paint black first then white (or your preferred color) over the black. Then when you sand to make it look distressed the black shows through and not the natural wood color. I love this and think it is super cute. Good luck!

How big is this? Do you have theme yet for the nursery?

I would distress it and use it to hang cute things on it

What are the dimensions? Is that a mirror on the side?

What about adding a backing, then making it into a babies 'busy box'?

I would leave the color wood to use as they grow. Now, to hold diapers in the bottom and cute things on top hanging. Also , you could use it to hand your lotions, diaper rash meds , or any thing you need in it. ( just tie them with pretty ribbon. ) Easy access! Its portable and can be brought around to another room if needed. Future use : as a jewlery hanging items as they grow older. Or even hair items. Think out of the box! Keep it you can use for lots of things.

The shape lends itself well to being turned into an animal shape. You add a face on the one side and paint it to finish off the look. An Elephant, a Giraffe, a Tiger... lots of possibilities.