You can publish your level once you have been able to complete the level yourself.
This means that in Racing mode you need to be able to drive to the goal without dying, and in the Adventure mode you need to collect all the map pieces, and then drive to the goal without dying.
After completing the level you get an option to "GO LIVE!". But remember, the better your level is, the more likely tou are of getting Likes from other players, and that means more coins! So keep editing the level until you are very happy with it, and you feel like it's the best it can ever be :)
You can share your level in social media, chat apps or other apps with the Share feature. Just go to your published levels list, and tap any level. You can then see a blue Share icon. Just tap it, and choose where to share the URL link.
People can then open the link with their mobile devices or computers. If they open the link using a mobile device AND they have Big Bang Racing installed, the game will automatically open your level! If they don't have the game installed, or they use a computer, then the game's Store page will open.