Walkthrough: After Klepto steal Cappy, quickly head across the bridge. You should be able to find a set of stairs to the right after you do, right between two different platforms. Reach the raised platforms of the two and ground-pound it. This will raise the other platform, allowing you to head to the next level of area. Once up on the next level, dodge the pesky bugs and make your way across even more poisonous rivers. Come to another big platform, ground-pounding the far right platform to form another set of stairs that lead to Swamp Hill.
Once on Swamp Hill, you will encounter two birds known as Tropical Wigglers that will attempt to block the path leading to Cappy. Wait for just the right time to jump past them, eventually reaching Klepto afterwards. Klepto will be on the top of another raised platform, one exactly like the ones you encountered on your way over to the bird. By chasing Klepto up to the very top of the platforms, he'll avoid you and head down to the lower platform. After he does, ground-pound the higher platform to easily defeat the bird and save Cappy.
After Klepto has been defeated, you will be free to roam Lost Kingdom. You will need 10 Power Moons to leave this kingdom, but since it's rather small and does have 35 Power Moons available to be found, it shouldn't take you too long to accomplish this. After obtaining a mere 10 Power Moons, you will restore the Odyssey Airship and head off to Metro Kingdom.