Unzip the file. If the unzipped file shows the messages individually as eml files the import-export addon and easily import them.

If I'm not mistaken the files have this format 'COM,S=15843,W=16400_2,SA'. They all have the same format but with different values for each file. Or at least this is what windows is showing under the 'Type' column. Am I looking at the wrong place? The files are located at '.[email_address]>.Sent.cur'

Did the unzip create a separate file for each message? Or is it an mbox format? I have no idea what format Hostgator uses.

It created a 'mail' folder. Inside the folder there are several folders which have the same name as the email addresses I had on hostgater as well as New, tmp, .Sent, .Draft, .Junk, .Trash, Cur, .Archive as well as some other files like dovecot.index, dovecot.index.cache, mailbox_format.cpanel, dovecot_quota, etc And each folder represententing an email address has the same files New, tmp, .Sent, .Draft, .Junk, .Trash, Cur, .Archive as well as some other files like dovecot.index, dovecot.index.cache, mailbox_format.cpanel, dovecot_quota, etc.

Ok, so our provider has given you a direct backup of the data stored in the Dovecote email server platform https://www.dovecot.org/. It is by and large not compatible with Thunderbird and is meant to be imported into a standing installation of dovecote.

You might get the import export tools to import the data if it is stored in EML or MBOX formats as David suggested, but it is up to you to identify which formats are in use. Perhaps start by referencing the dovecote documentation on file formats. https://doc.dovecot.org/admin_manual/mailbox_formats/

Thunderbird can access a dovecote server as an IMAP client and get mail, but it has no understanding of the data structures dovecote uses, nor a direct import facility from Dovecote.