How does civic work? I have been doing a lot of reading up on bitcoin, but I can't find anything that explains how some technologies are used in conjunction with the blockchain.

From my understanding, the reason bitcoin is distributed is the blockchain is stored at multiple nodes (miners). When they validate transactions they get added to the blockchain and miners get a reward/transaction fee.

My current question is around civic. What would be the benefit to miners when it comes to a technology like civic (i.e. what would they mine and why)? What is the engine that keeps this data distributed?
Firstly, the 'engine' to keep civic decentralised is bitcoin itself. Its called PKI and anyone can generate as many private-public keypairs as required for free. These keypairs are already used for authentication in applications all around.

The simplest app one can think of is sign something and put it on the blockchain, just so someone knows you are alive. Civic is the app that does just that - helps you put data on blockchain and notifies any interested parties that your phone is still alive. 

Secondly, there is no other part. Bitcoin is the 'engine', cypherpunks intended people to assemble the 'car' themselves. But in real-life some prefer to pay for the DIY instruction manual written in ELI5 manner or simply don't even know what they are buying.