You will get new level once you have a certain Experience number. You can gain Experience by performing different activities in Clash of Clans such as: Destroying Town Hall while raiding, upgrading your Buildings, donating troops for your Clanmates,…
In game, you can see you current level and current experience on the top left of the screen. While visiting other players village, you can’t see their current experience, just the level.
Thank RyanSayHi for the screenshot :D.
Every level before 200, you need more 50 experience than the previous level to increase your level. From level 200, the number of experience you need to reach new level increases by 500. Till now, the maximum level in Clash of Clans is 300.