Once you have completed the first arena by fighting and gaining Trophies, you will be automatically promoted to the next one. You will also be rewarded with a new Dragon which is useful for the next Arena… and the new Dragons have really cool special attacks!! Promoting between arenas of the same kind (Element) does not reward a Dragon.
To see how many Trophies you need to gain to advance please see the Arenas Tab inside the Arena.
You can be demoted to the lower Arena if you lose your battles! Don't worry, you won't lose your Dragons... they will just need a little rest. ;)
Please note - You only get the Arena Dragon rewards ONCE (the first time you are promoted to a new Arena) and if you're demoted and promoted again, you don't get the reward a second time.