In the AcyMailing configuration, tab Mail settings, you can choose this sending method.
In order to send with Mailgun, you will need a Mailgun account and to create a domain.
In your Mailgun account, access the "Sending" part and create a domain if not done. Copy the domain to put it in the AcyMailing configuration along with the region of the domain (US or EU).
Edit the domain and access the API keys section to create and copy the Private API key that you can add in the AcyMailing configuration. You can also access this in the Settings > API keys menu.
Once you have added all information in the AcyMailing confiugration (region, domain and API key), you can test them thanks the the "Test credentials" button.
Don't forget to save the configuration!