To nestle into something is to get snug, comfy, and warm in it. You might nestle into your mother’s shoulder, or into a pile of blankets on the couch.
People must think that birds live comfortable lives, because they describe really cozy, safe, and comfortable places as nests: places to nestle into. And it’s not just people and birds that nestle. A cottage might be nestled into the crook of a hill, or a mouse nestled into a pile of wood shavings.