Core pg 266, Rigging and Limits
When you’re jumped into a vehicle, drone, or other device, the limits of that device are increased by the rating of your control rig. This includes vehicle and drone Sensor, Speed, and Handling, and the Accuracy of mounted weapons when used by the rigger.
Speed and Handling aren't part of dice pools anyway. They're just used as the Limit for other rolls. So when you roll Reaction 6 + Pilot Ground 6 with Handling 3 means that out of those 12 dice you can only count 3 successes towards the drive check. But you're using a Rating 1 Rig so the Handling goes up to 4. Now you can count 4 successes.
Core pg 452, Control Rig
When you’re jumped into a vehicle or drone, the control rig provides its Rating as a dice pool bonus on all Vehicle skill tests. Additionally, the rating of your control rig is added to the Handling and Speed of any vehicle you are jumped into. As if that was not enough, your Vehicle Test thresholds are reduced by the rating of your control rig (to a minimum of 1), again when you’re jumped in.
Here they're actually detailing 3 separate effects.
1) Add the Rating to Vehicle skill dice pools. That means any Pilot Craft checks, and also any Gunnery checks. So now that check from before is Reaction 6 + Pilot Craft 6 + Rig 1 = 13 dice.
2) It repeats what we knew already, that the Rating is added to the Limiting factors. So again, where your vehicle has Handling 3, the Rating 1 Rig brings it up to 4.
3) And the Rating is used to reduce the Threshold for Vehicle Tests. Meaning when the chart on Core pg 199 says that a hairpin turn has a Threshold of 3 (you need to roll at least 3 hits to succeed), your Rating 1 Rig brings it down to 2.
So... you're rolling Reaction 6 + Pilot Craft 6 + Rig 1 [Handling 3+1=4] hoping to roll at least (3-1=2) hits to succeed at the maneuver.