Connect from Technology. Choosing Facebook opens a Meta sign-in window; approve the request there and the window closes on its own.
You must be a Page admin
Not an editor, not a moderator. Admin of the Page.
Being an admin of the business is not the same thing, and that single distinction causes most of the connection failures worth reporting. The sign-in succeeds, Meta returns nothing, and no Page appears to choose.
Page roles are changed in Meta's own tools, not here — see Meta's page on roles and permissions, linked below.
If you admin more than one Page
You are asked which one. The choice is per connection, so a business with a main Page and a second location Page connects the one it wants to post from.
What connecting gives you
Posting to the Page, and the reach and engagement figures behind it — see reach and engagement.
Instagram is a separate connection
Even though it lives behind the same Meta account. See Instagram, which has a requirement of its own.
What you are handing over
Permission to post to the Page, and to read the Page's own reach and engagement figures. Nothing else on the account, and nothing on a personal profile.
Your Facebook password is never given to Omnyra. You sign in to Meta, Meta asks whether to allow the connection, and Meta reports the answer back. Approval can be withdrawn from Meta's settings at any time, independently of anything here, and disconnecting in Omnyra does the same from this side.
If it stops working later
Meta expires the permission periodically, and a connection that worked for months can simply stop. That is the ordinary case and it is fixed by connecting again — see a connection stopped.