You’re responsible for securely sharing that URL using a communication channel of your choice like in an encrypted chat message, or using something less secure like unencrypted e-mail, depending on your threat model.
The people you send the URL to then copy and paste it into their Tor Browser to access the OnionShare service.
If you run OnionShare on your laptop to send someone files, and then suspend it before the files are sent, the service will not be available until your laptop is unsuspended and on the Internet again. OnionShare works best when working with people in real-time.
Because your own computer is the web server, no third party can access anything that happens in OnionShare, not even the developers of OnionShare. It’s completely private. And because OnionShare is based on Tor onion services too, it also protects your anonymity.