Only mailbox owners (formerly known as delegates) can access a shared mailbox. Owners can see the contents of the mailbox and send email from it.
A mailbox sponsor can manage the shared mailbox. That is, they control who the mailbox owners are, are notified before the shared mailbox goes to deprovisioning and can extend the expiration date for it.
So if you are a sponsor of a shared mailbox and aren’t already an owner you can make yourself one by following the instructions.
How does an owner access the mailbox?
Please see this Microsoft support page.
Why aren’t sponsors automatically owners?
Because many times they do not want to be. Sponsorship is often at a higher organization level than the utilization of the shared mailbox and the sponsor does not want to have their email cluttered by including it. If you are an exception? Great! Because as sponsor you already have the ability to grant access to yourself — or whoever you want to delegate such access to.