To copy a public mod made on LearnToMod, go the blockly editor page for the mod, then from the dropdown menu at the top of the screen select 'copy'. Now you'll need to create a new mod. Go to http://mod.learntomod.com/programs and create a new Blockly multiplayer mod. Then on the editor page for that mod, select 'paste' from the dropdown menu at the top of the page.
Afterwards, click on the editor screen to make sure its saved. Then press the mod button under the editor screen. That will add it to your mod chest.