Whenever a conversation or a task gets assigned to you, it appears in both your Inbox and Assigned to me mailboxes. So it’s perfectly possible to work without using Assigned to me at all, only your Inbox.
However some people like to keep their Inbox very clean, a practice referred to as the Inbox Zero method. In this workflow, conversations usually remain in the Inbox for a very short time. As soon as a conversation is read, it will be either archived, snoozed or labeled to be worked on later.
When users of this technique are assigned to a conversation, they can see it in their Inbox, acknowledge and archive it, knowing it still remains in Assigned to me. That’s when this mailbox becomes useful.
To remove a conversation from Assigned to me, it must either be assigned to someone else or marked as closed. See this guide to better understand when to archive and when to close a conversation.