I know that deploying flaps will increase both drag and lift, thus increasing L/D ratio. But how exactly does that correlate with the increase/decrease of the rate-of-descent?
According to the FAA, lowering or extending flaps allows one to increase drag without an increase in speed. This increases descent rate for a given speed.
One must be careful to consider what factors are held constant to evaluate the effect. For example, one can lower flaps during level flight. This will result in a lower pitch attitude--since they produce more lift, less angle of attack is required. If power/thrust is not increased, speed will decrease due to increased drag.
However, an aircraft flying at a constant speed will descend faster (higher rate of descent) if the flaps are extended compared to when they are not.
Flaps reduce L/D. The lift increase is not as much as the drag increase, and ratio of lift over drag goes down. So if L/D flaps up is 12 and I am descending in a glide at X knots with a vertical speed of Y (Y being 1/12th of my forward speed), and I deploy flaps and L/D is now 8, and I'm still at X knots, vertical speed will be higher, with the glide angle now much steeper at 8:1 vs 12:1 (vertical speed will have increased from 1/12th to 1/8th of my forward speed). I can reduce the descent rate, but for that I have to slow the airplane down but the new vertical descent rate rate will still be 1/8th of my reduced forward speed, which will generally still be higher than it was flaps up.
If your hold your Speed and Angle of Attack constant and deploy your flaps, you will gain Altitude. That means, that you can reduce your Speed OR your Angle of attack (or both), while maintaining your altitude constant.
So depending on what situation, Flaps does not have to increase/decrease your rate-of-descent. It is more an interaction between Flaps, angle of Attack and Speed.