Airmail uses iCloud Drive for two purposes, of course, only if you allow Airmail to use it.
1. iCloud Sync
Airmail syncs its preferences and accounts over iCloud to multiple devices. This helps you keep the app in sync over all the devices and eventually enhancing the experience while using Airmail on all platforms. This can only be done if you allow Airmail to use your iCloud Drive. You can find the detailed article about iCloud here.
2. Save Messages on iCloud Drive
Airmail now offers you the option of saving the important emails (in PDF form) and attachments to iCloud Drive. You can find the detailed article about saving messages to iCloud Drive here.
How to Enable iCloud Drive for Airmail?
If you do not have iCloud Drive enabled for Airmail and try to use an iCloud dependent feature in Airmail, you can get some errors while doing it, for example, "Verify your iCloud Credentials" or iCloud "Disabled by System".
To enable iCloud Drive for Airmail, go to iPhone Settings > iCloud > iCloud Drive and you should get to the list as shown in the picture below.
This is the primary setting which should be enabled if you wish to use iCloud with Airmail.