The way that MemberMouse functions is that if an account is expired the customer will be unable to login and instead will be redirected to an error page.
By default, they will be shown an error message saying their account is expired. However, this error page is simply a WordPress page so you can customize what's shown to the member based on their account being expired, providing you with the opportunity to display specific instructions for what they should do. This article provides more details on how to do this: Create Custom Error Message on Error Core Page
The easiest way for expired members to reactivate their account is to simply sign up again via a purchase link (Create a Purchase Link) using the same email address and password as their initial account. This purchase link can be included in the custom error message or sent to them in an email.
This is also the method for reinstating a member who is in 'pending cancellation' status.