You only do this once. The first time you open MoveIt, macOS asks you to confirm it. Here's how.
In the MoveIt window, drag the MoveIt icon onto the Applications folder.

Open MoveIt from your Applications folder. macOS says it could not verify MoveIt. Click Done, not Move to Trash.

Choose Apple menu › System Settings…, or click System Settings in the Dock.

It's near the bottom of the sidebar. You can also type privacy in the search box at the top.

Under Security you'll see "MoveIt was blocked to protect your Mac". Click Open Anyway.

Use Touch ID or enter your Mac's password.

MoveIt opens and lives in the menu bar, at the top right of your screen. macOS won't ask again. On the welcome card, click Start Free Trial; it needs the internet once. Bought MoveIt Pro already? Click Redeem Key instead.

macOS checks apps for a certificate from Apple. MoveIt doesn't have one yet, so macOS can't check it and asks you to confirm it once. MoveIt has no account and never uploads your files.
On macOS 14 Sonoma or earlier there's a shortcut: right-click MoveIt in Applications, choose Open, then click Open.
Start dragging any file and give the mouse a quick shake. A shelf appears: drop the file on it, go wherever you like, and drag it out again.
Opens an empty shelf.
Takes a screenshot straight onto a shelf. macOS asks for Screen Recording permission the first time.
The button on every shelf, with all the tools: resize, convert, QR code, ZIP and more.