If you don't see the achievements on your profile screen, go to "Trips" and try to pull down the list with your flights to update. Wait for the flights to sync, restart the app and check for badges. Below you will find screenshots with explanations.
Go to your "Trips" → Swipe down to update your account information.
Wait until the "Syncing" message disappears.
Go back to your Profile → you should see all your badges.
Also, there are different types of levels, as you can see under the icons, there are numbers. For example, planes with 3 numbers mean that you are at level 3 (every 5 planes changes differently).
Go to the Trips screen → Swipe down to refresh.
Then go back to your Profile → Scroll down, and you will see your achievements again.