The iliopsoas (psoas for short) is the primary muscle that lifts the legs above 90 degrees to the front and side. The iliopsoas is a hip flexor that works with the quads to begin lifting the leg before taking over above the 90-degree point.
The quads are strongly engaged throughout an extension, but don't worry about that. They should be engaged whenever you are flexing at the hip. The myth that you should "lift your leg from underneath" is still floating around, and I'm here to unequivocally state: It is anatomically impossible to use the muscles on the backside of the leg to lift the leg up.
Here is how to strengthen the iliopsoas for higher extensions.
Start by sitting on the front edge of a chair, keeping equal weight on both sits bones. Without shifting in any direction, lift one knee up as high as possible. Lower your leg to touch just the toe to the ground.
Repeat this 15–25 times. It won't take very long before you'll experience that deep "tired" feeling in the front of the hip. To increase the challenge, you can wrap a TheraBand around your legs above both knees and use the resistance to increase the work of the iliopsoas.
After strengthening both sides, be sure to stretch out and release both the quads and iliopsoas muscles.