It's related to a productivity strategy called "inbox zero" where you immediately deal with every email in your inbox when you check your mail. "Deal with" might be to delete it if it's junk, respond immediately if you can, or defer it to later. A lot of these clients are aimed at making this process as fast and efficient as possible. So taking actions on an email are often just a swipe, and there are facilities to defer an email to a particular time, or adding it to a to-do list.
It's mainly intended at people who get too much email.
I use Outlook. The thing I like about it is that it integrates my gmail and icloud accounts nicely. Native iOS app doesn't work well with GMail because you won't get notifications for new mail. And the GMail app doesn't work well with my icloud.com email - or iCloud doesn't forward to GMail reliably.
So I use the MS Outlook app and both are integrated seamlessly.
I didn't use Mailbox, but I think one of the key features is the snooze button, that makes an email disappear from your inbox and then reappear later. I think at the time Mailbox came out Google didn't offer that feature. The GMail webapp still doesn't have it but now Google Inbox does.
It seems like Mailbox had a loyal following but I guess my email use is just not sophisticated enough to understand.