As far as I can tell, the UniFi APs don’t come with a SFTP server, so you’ll need to use SCP (which operates over SSH).
Unfortunately, it seems that most developers have dropped true SCP support from their software in favor of SFTP. Luckily, if you are on a *nix system (including MacOS) you can use the utility scp to accomplish this task.
scp nameofuser@ipofap:../../var/log/messages /local/path/on/computer
For those running Windows, you can either setup a Linux VM or use Microsoft’s WSL to gain access to SCP.
Note that the ../../ are important. By default SSHing into a UAP lands you in a folder /etc/persistent, you need to get back up to the root and then navigate into the /var/log where we’ll grab the messages file.