While it's not currently possible to download or export individual mods made on LearnToMod, there are some options.

You can export a .zip backup file of your LearnToMod world and play it on a non-LearnToMod server-- it still must be played on a compatible Java minecraft launcher,  on a 1.9 or 1.9.2 profile, and on a multiplayer server.  (Adventure mode isn't supported.) The backup can be useful as a way to save anything that's permanently built on the world like structures, changed textures, and if there are any mods that are permanently running on the world, they might be included in the backup as well. 

To do this, click the My Stuff button in the middle of the LearnToMod homepage, scroll down to worlds -- select a world. The newest is at the top of the list. Then scroll down and click the .zip backup button. (It's not possible to get a .jar file)

How to Copy and Paste Code

Each badge or editor screen has a small gray JS button that allows you to view the code in JavaScript format. (located near the Simulator button). To copy the code, you'll have to do so from the freeplay modding section. At the top of the page of the freeplay modding section there's a drop down menu -- select copy. Then you can paste the code in a plain .txt document. What you do with it next is up to you.