MQTT is a lightweight standardized communications protocol. MQTT is how the switch communicates state to HA and how HA sends commands to the SONOFF. The default SONOFF firmware doesn’t integrate with HA. What you’re flashing is a replacement that does.
MQTT is supported by most, if not all home controllers (Openhab, Domitz, Homegenie etc). There are also clients for android, iphone and other operating systems that can be used to control switches.
The authors of the new firmware could have written something that enabled a direct connection, but it would unlikely to be as widely used as MQTT.