The hip flexor muscles work on the front of the hip helping to bend it forwards. If the torso is held stable then the hip flexors can be used to lift the leg forwards. If the leg is stable then they can be used to tilt the pelvis forwards relative to the leg.
Depending on whether the lumbar spine is kept stiff or allowed to bend this can result in the whole torso moving towards the leg or simply an increase in the backward curve of the lumbar spine.
One way to strengthen the hip flexors is to practice lifting one knee to the chest while balancing on one leg.
Initially you can focus on lifting the knee while keeping it bent. So that your hip flexors have a stable foundation from which to act on the thigh and knee, you can bend your standing knee slightly and even turn the foot out. Make the standing leg hip feel strong. Then work at flexing the hip of the leg you are lifting.
One idea for strengthening muscles is to give them room to contract. Standing on one leg, one way you can do this for the hip flexors is to tip your pelvis backwards as you lift the knee. Keep your ribcage upright so that you flatten your lumbar spine as you tilt your pelvis backwards. This may then give the psoas extra room to contract near the top of the action.
Two options for a straight knee hip flex while balancing on one leg are to straighten the knee after lifting it or to lift the leg with the knee straight.
When straightening the knee you may find it helpful to make the knee strong. This may make it easier to use the tensor fasciae latae and sartorius as accessory hip flexors.
If lifting the leg with the knee straight, start with small lifts. Then work at gradually lifting the leg higher.
I generally have my students do all exercises slowly and smoothly. I'd suggest you do the same.
In a lunging position with one leg forwards and the other leg back you can strengthen the hip flexor of the back leg by straightening the knee and pushing the foot strongly into the floor.
Hands can be on the floor or lifted.
Work to prevent your pelvis from lifting as you do so.
It may feel like you are pulling your feet towards each other as you do this.
Here again work at a slow and smooth contraction and a slow and smooth relaxation.
In a standing forward bend like front triangle you can strengthen you hip flexors by creating an upward pull on your front leg. If you keep the knee straight while doing so you may find that your quadriceps engage but more importantly you should also feel some tension at the front of your hip as your superficial hip flexors engage (in this case, rectus femoris, but also possibly tensor fascia latae and sartorius).
One of my favorite hip flexor strengthening exercises is also a challenging balance exercise.
Doing a standing forward bend, shift your weight to one leg and lift the other leg. But rather than reaching the lifted leg back (like in Warrior 3) pull it forwards while keeping both knees straight.
This exercise is a little bit easier with the hands on the floor. It becomes a fun challenge when you lift the hands (or try to.)
You can also use plank pose to strengthen your hip flexors.
To get used to using your hip flexors in this pose start in cat pose (on all fours) with the elbows straight and the knees beneath the hips. Straighten one leg and place the foot on the floor.
Focusing on the foot of the straight leg, press it strongly into the floor. Press it down strongly enough that you can easily lift the other knee off of the floor. Pause, then lower. Repeat on the same side or switch sides each time.
Rather than suddenly pressing the foot down and suddenly lifting the leg, try to gradually press the foot into the floor. Feel your hip flexors activating (the straight leg hip flexors). Then gradually relax the pressure of the bent knee on the floor. Then lift it.
Once you get the feel for your hip flexors, try the plank with both knees straight. Focus on pushing both feet strongly into the floor.
Boat pose can also be used as a hip flexor strengthening exercise. Sitting upright, lean back and balance on your buttocks. Keep your feet lightly touching the floor with the knees bent and your arms reaching forwards. Lift one knee. Pause then lower the foot and switch legs.
You can repeat this a few times and then rest (or do a hip extensor strengthening exercise like yoga table pose or bridge). Then repeat the exercise but this time straightening the knee after you lift it.
You can also try the same two hip strengthening exercises but lifting both legs at the same time.
Seated Wide Leg Side Bend can also be used as a hip flexor strengthening exercise. Bending to one side, focus on pulling up on the leg you are leaning towards. Keep the knee straight as you do so. While it may seem strange to pull the leg up in this position, think of it as a simple way of activating your quadriceps and your hip flexors. The feeling is almost the same as doing a standing leg lift with the knee straight, the only difference being that you are sitting on the floor while doing so.
And instead of the leg moving upwards, what you should find is that you actually pull your torso towards your leg.
You can use the same hip flexor strengthening action in Hurdlers stretch. Focusing on the straight leg first while sitting up, for practice, pull up on the leg while keeping the knee straight.
You can use it as a breathing exercise. Inhale as you pull up on the leg. Exhale as you relax the pull.
Repeat the same actions while bending forwards. Here again you should notice that hip flexor activation helps you to pull your chest towards your straight leg.
As a preparation for a exercising your hip flexors in seated forward bend, first sit upright and practice creating an upward pullon your legs one at a time.
If you have trouble with the idea of pulling up on your leg, or you have difficulty activating your quads and hip flexors in general, the Basic Muscle Control for Quads, Hip Flexors, Hamstrings and Glutes video is designed to help. It includes the basic exercises plus the "no-fail" exercises that I resort to for teaching my students to feel and control these muscles.
The ebook Muscle Control for Better Flexibility includes basic exercises for learning to feel and control these muscles in the context of a routine. It also includes an optional video of the routine.