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It's easy to keep your legs straight when you're standing up or laying flat on your back, but as soon as you try to sit up or bend down, you need to bend your knees as well as your waist. (Try it. Lay down flat on the floor, then try to sit up straight while keeping the backs of your knees pressed against the ground. You'll have trouble long before you reach 90 degrees.)
Why is this?
First off, I would be careful generalizing people's experiences with this exercise (..."Try it...You'll have trouble long before you reach 90 degrees"...):
Source: Yoga Tune Up
The muscles you are feeling at 90o, and the lady pictured above perhaps not even at 180o are the hamstrings. They run along the back of your thigh, attaching at both the hip and the knee. They flex the knee and extend (straighten) the hips.
Source: Precision Nutrition
When you do the exercise you are describing, the pelvis starts pulling on the hamstrings from the top because your knees are staying in place. Bending the knees releases the stretch.
