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. Keep Finder for those jobs, or compare full file browsers against the specific features you need.
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 / macOS | 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, or copy then Option-Command-V | Drag into compatible apps or Finder folders |
| 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 files while you find where they go | Copy a selection to the Clipboard | A visible shelf for several separate drops |
| Show recent downloads without opening a window | Downloads stack in the Dock | Yes, and any other folder you name |
| Reach files while another app is in front | Dock stacks or the Clipboard; browsing brings Finder forward | The shelf draws over the current app |
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.
You can also use the Clipboard: copy a file, find its destination, then paste it, or use Option-Command-V to move it in Finder. Apple documents those keyboard shortcuts. A shelf gives you a visible place for several separate drops, without replacing the Clipboard. For a drag across windows, you can 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.