Flexion means "to bend" while contracture refers to the soft tissues being 'fixed' or 'stuck.' In more functional terms, a flexion contracture means your knee is flexed or bent and can't straighten all the way.
When a joint stays in one position without moving for a long enough period of time, the tissues start to stick together. Fibrous tissue fills in between the soft tissue fibers. Loss of motion means the joint can no longer move the way it's supposed to.
The usual result is pain and loss of function. It becomes harder and harder to climb stairs, get up and down off the floor, or even stand for long period of time.
Surgery to release or cut the contracted tissue is sometimes advised. More conservative treatment involves stretching and other exercises.