This must be something newer then because as early as a month ago I could find no way to do this. That may be failing of their documentation however I clicked every button/icon there was in the mobile app and found no way to do it. I can see now that by clicking the icon to the right of the folder name I can choose to have it "Auto-sync" which I assume means the data is stored on my phone but it's still confusing.

He doesn't want to selectively sync only certain files with his phone. He wants to browse the available files and then on demand download that file. This is a common use case for me also and something that dropbox supports pretty well.

That's what it does.