Why exactly is SATA Air Açores required to provide this service?
According to this official announcement by the Regional Government of the Azores, a new "Air Passenger Transport Model" entered into force on March 29, 2015.
This Air Passenger Transport Model creates a "routing service", described as:
"a service, free of charge, for passengers travelling within the Autonomous Region of the Azores, to/from Mainland Portugal or Funchal, who wish to use any of the gateways in the Autonomous Region of the Azores with destination to any other island of the archipelago."
The announcement also states:
"The provider of this service is the concessionaire of public inter-island air services, SATA Air Açores, SA."
Who pays for those flights?
Again according to the announcement:
"Following the negotiations between the National Government and the Regional Government of the Azores [...]"
The Regional Government of the Azores and the Portuguese National Government negotiated the Air Passenger Transport Model, so they are probably the ones paying SATA for each passenger flown for free.
What's the motivation for this?
The announcement does not clearly state the motivation, however it sets the following rule:
"The travel route, from arrival to the gateway to the departure of the connecting flight, cannot exceed 24 hours."
Most tourists arrive in the Azores by its main international airport, João Paulo II Airport on São Miguel Island. As I see it, the government's goal is to encourage tourists to take a connecting flight to another island when they arrive in the Azores. If it costs more to fly to Flores Island than it costs to fly to São Miguel Island, many tourists won't pay extra to reach Flores. With a free flight, they are way more likely to compare the islands and choose the one they prefer. Thus islands with less international connections than São Miguel can benefit from tourism as well.
It is also interesting to note that along with the "routing service", the Air Passenger Transport Model creates a "Social Mobility Aid". It is described by the government announcement as:
"the reimbursement paid by the State to resident passengers, citizens with resident status and students who use the regular air services between Mainland Portugal and the Autonomous Region of the Azores or between the Autonomous Region of the Azores and the Autonomous Region of Madeira."
Overall the government seems to want to improve the connection of its autonomous regions by developing inter-island connections as well as connections between islands and mainland Portugal.