How do fireplace direct ignition control modules work?
The voltage at the igniter is probably high enough to damage your multimeter, if you succeed in getting it to operate. 

The high voltage transformer output (the tab on the black cylinder) goes to the electrode not to the mains.

You should not be fooling with this safety-critical subsystem, please cease before you electrocute yourself or blow up your house.  



As to how it works, the incoming mains is stepped down by the transformer T1 and rectified and filtered, then likely regulated down to 5V to power the microcontroller U1 which controls the ignition sequence and gas valve actuation via the Omron relay, including flame detection, limited restart attempts and safety lockout. The safety redundant components near the connector are likely a flame rod type detection scheme. It’s very important that the gas not flow for more than a very brief time without a flame, lest the gas fill your house and eventually find a spark with unfortunate consequences.

The spark itself is typically generated by triggering an SCR to discharge a capacitor into the primary of transformer T2, creating a high voltage spark at double mains frequency, and not isolated from mains (so an electrocution hazard).

The board has redundant parts for safety, the micro has safety-critical approved hardware and firmware and the circuit is conformally coated. 

It is designed to fail only in a safe manner (to sufficiently high probability to satisfy UL and CSA that it won’t kill too many people), which it apparently has.