I have 5 GB of stuff in
~/Library/Application Support/iLifeAssetManagement and I'd like the space back. What is this stuff and how can I reduce its footprint?
Yes you can delete it. I would make a backup of your Mac before deleting anything in ~/Library just in case you run into corruption.
This folder contains iCloud synced photos from your PhotoStream, so you might just have the folder fill up again unless you want to poke at that folder and make sure you turn off iCloud syncing before you do the pruning.
Deleting things from the iCloud photo stream directly will reduce the size footprint on all your devices so that's what I'd recommend in this particular case. You can sort things by size in Finder, so that might help you see if just a few large videos are responsible or if you instead have hundreds of small photos all adding up.