Slimy Leaf blocks generate as part of Slimy Trees. They drop Slimy Saplings and Gelatinous Slimes occasionally when broken or they decay. The leaves will begin to decay when updated and over 4 blocks away from a Congealed Green Slime block or a Congealed Blue Slime block.
The fastest tool to break the Slimy Leaf with is the Sword. They can be collected and placed using Shears or a Silk Touch modified tool.