What is this turbine engine panel that opens on landing?
What you saw is the thrust reverser of the engine, which redirects some of the airflow forwards and therefore helps slowing the aircraft down. The grid like structure are the cascade vanes. This is what it looks like inside the engine:

(Airbus A380 FCOM - Engines - Thrust Reverser System)
The blue arrows indicate the flow of the so called bypass air, which is air that goes through the fan (and is therefore accelerated), but does not go into the core engine. With the thrust reverser deployed, this air is redirected forwards. The core air going through the actual turbine engine will still exit at the back.
Since there is more bypass air than core air1, there is a net thrust generated in the forwards direction.
If you are wondering what the pilots are doing to engage reverse thrust, this is a picture of the A320 family thrust levers:

(image source)
The main thrust levers are pulled all the way back during the flare to the position shown in the image (labelled 0). To open the reverser doors, you pull up on the smaller handles at the front of the thrust levers. Once the reverser doors have opened, the thrust levers can be pulled backwards to add more reverse thrust (range labelled REV in image). To close the reverser doors, the thrust levers are moved back forwards and the smaller handles are pushed back down.
Note that on a dry runway reverse thrust does not slow the aircraft down faster (assuming auto brakes are used). The brakes on the main landing gear are capable of achieving the same deceleration without reverse thrust. Engaging reverse thrust therefore reduces brake temperatures and wear.
The thrust reverser doors can look different depending on the engine type. Based on your image, the aircraft you were on was equipped with IAE V2500 engines2. The reverser doors on the CFM56 engine, as used on some A320 family aircraft, look a bit different3:

(image source: Wikimedia)

1 The IAE V2522-A5 engines used on the Airbus A319 have a bypass ratio of 4.9 : 1 (source).
2 FlightAware shows a EWG4063 flight tonight from Munich to Dortmund and according to planespotters.net Eurowings uses IAE engines on their A319s.
3 The CFM56 engine is also used on the 737, but uses the cascade thrust reversers there (thanks to DeepSpace for pointing this out). See also Why is the CFM56 produced with two different thrust reverse options?