I have been reading a lot on flexure bearings, but it's still not really clear what they are. From everything I've seen, it seems like a flexure bearing is just a spring.
Can someone give me a simple explanation of what they are and their use?
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From the Wikipedia article you linked,
A flexure bearing is a bearing which allows motion by bending a load element.
So it's something that provides support, but also allows motion by bending a component (as opposed to sliding).
There are several examples on the Wikipedia page, but I would add (in addition to those examples) that probably the most common flexure bearing would be the vibration mount.
It is typically a piece of rubber that is designed to provide support, and it allows some relative motion by bending.
I have vibration mounts holding the engine and transmission in my car, holding the ball joints, sway bars, and other suspension components. They're in the blender in my kitchen, in my washing machine, etc.
You could consider any rubber foot on a component designed to prevent rattle/hum (refrigerators, etc.) as vibration mounts and thus flexure bearings. They all provide support and allow motion by bending.