The honest version first
If you want something that browses your disk, searches it, renames things in bulk and shows a column view, Tray is not it and will disappoint you. Look at Path Finder or ForkLift — both are real Finder replacements and neither is trying to be this.
Tray is for the other half of what Finder gets used for: a shelf that holds the handful of files currently in play, reachable from inside whatever app you are already in.
Side by side
| Task | Finder | Tray |
|---|---|---|
| Browse a whole disk | Yes, that is the job | No. It holds what you put on it |
| Search everything | Spotlight, and it is good | No |
| Rename a file | Yes | No — a pile can be named, a file cannot |
| Preview a file | Space, in a window | Space, from the shelf |
| Move a file somewhere | Drag between windows | Drag off the shelf into anything |
| Make a folder from a selection | Right-click → New Folder with Selection | Put in a Folder…, copying or moving |
| Send to the Bin | Yes | Yes |
| Share by Mail or AirDrop | Yes | Yes, from the card |
| Hold a file while you find where it goes | No. A window has to stay open | That is the whole point |
| Show recent downloads without opening a window | No | Yes, and any other folder you name |
| Reach it without leaving the app you are in | No — Finder comes forward | Yes, it draws over the top |
What changes in practice
The Finder window you keep open at the side, the one showing Downloads, is the one this is for. It is open because files arrive there and you need them a few minutes later. A shelf does that without a window: recent downloads are already on it, and so is any other folder you point it at — see watching a folder.
The second thing is the drag that needs two windows at once. Finder cannot help with it; you need both on screen or you need to hold the drag through Mission Control. A shelf lets you put the file down and come back — the four ways to do that.
What it does to your files
Nothing, unless asked. A card is a reference to a file where it already lives, so putting one on the shelf does not copy or move it, and taking it off again leaves it exactly where it was. The two exceptions are both menu items you have to go and find: Put in a Folder…, which asks where and whether to copy or move, and Move to Bin.