You come back to the page you were trying to open. Every address in the project editor is behind a sign-in check, so opening one while signed out sends you to the sign-in page, and signing in returns you to the editor step you asked for rather than to the home page.
What is stored, and for how long
The check keeps the path you tried to open in two places: on the sign-in address itself, and in your browser's storage. That stored return address is good for 30 minutes. Sign in inside that window and you land where you meant to go. Leave it longer and the address is discarded, and you finish on your account settings page instead.
Registering works the same way. If you do not have an account yet, the return address survives the move from the sign-in page to the sign-up page, so a new account also lands back in the editor.
Only the path is kept
What is stored is the path, not the whole address. Anything after a question mark is dropped, so a link that carried extra parameters returns you to the plain editor step. In the builder that costs you nothing, because every step's address is a plain path.
Being signed in is not the same as being on the project
Passing the sign-in check gets you into the builder. It does not get you into a particular project. A project your account is not a team member of cannot be opened whether you are signed in or not, and the editor reports that as Error Loading Project rather than as a permission message.
So there are two different failures here, and they need different fixes. Bounced to sign-in means you have no session. An error on the editor page after signing in means the account you signed in with is not on that project's team, which only someone already on the project can put right.
If you are bounced over and over
Being returned to sign-in immediately after signing in usually means the session was not accepted rather than that it is missing. Why you were signed out unexpectedly covers that case.