Reciprocal inhibition is an often quoted term used with respect to stretching.
The idea is that if you activate one muscle its opposing muscle will automatically relax. It's a "built in" mechanism so that muscles don't work against each other.
If you want to relax one muscle, say the hamstrings, then you engage it's opposing muscles, say the quadriceps. Supposedly, engaging the quadriceps causes the hamstrings to relax through "reciprocal inhibition."
The only problem with this idea is that there are situations where opposing muscles actually do work against each other. They do so in order to stabilize a joint. Or they do so to give better control of a joint.
It's quite frequent to hear the instructions for reciprocal inhibition given while doing static stretching techniques and that's the main problem. Reciprocal inhibition is more likely to occur when the body is moving.
As an example, it may come into play if you are swinging or kicking your leg forwards. Because your leg is moving forwards the hamstrings will relax so as to not inhibit this action. And if you start with your knee bent and then engage the quads to straighten the knee as you kick your leg forwards, then the hamstrings may very well relax to prevent from being damaged.
However, if all you are doing is a standing forward bend while keeping your quads engaged, the hamstrings aren't going to automatically release just because your quads are engaged. Part of the reason is that the quads mainly work on the knee joint. Only one part of the quads crossed the hips and acts as a hip flexor, the rectus femoris! And pulling up on the knee isn't necessarily going to cause this part of the muscle to engage and cause hip flexion.
In a forward bend you'd probably want to activate the psoas and the illiacus and for those to activate I'd suggest focusing on tilting your pelvis forwards (and pulling forwards on the front of the lumbar spine at the same time so that the lumbar spine and pelvis tilt forwards as one unit!)
But back to reciprocal inhibition. If you want to use it in a "static" stretch like a standing forward bend, then it may help if you imagine that you are kicking your leg(s) forwards.
What happens when you kick your leg forwards normally? Your leg swings forwards and if you kick high enough it moves towards your chest. Imagine the same thing in a forward bend, even with both feet on the floor. Imagine kicking your legs towards your chest. This intention will cause your hip flexors (and perhaps your quads also) to engage and that in turn, via reciprocal inhibition, may cause your hamstrings to relax and stretch.
In any stretch where you are focused on relaxing the muscles you are stretching, then focus on activating their opposing muscles by trying to "muscle yourself" deeper into the stretch. Rather than just contracting the opposing muscles, contract them with the intent to move your limbs in the direction of stretch.
An alternative is to use a stretching technique like Active Isolated Stretching (AIS) where you do kick your leg to stretch the hamstrings. (This system has a whole repetoir of stretches, not just hamstring stretches, but stretches for the whole body.)
If you aren't sure how to deliberately activate muscles like your quads and hip flexors in a way that does help you improve flexibility, or how to teach it, Active Stretching and Muscle Control shows you how.