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There are actually a couple reasons for this - I'll try to explain as best I can in plain language (sometimes hard to do).
Chronic kidney disease frequently causes a chronic inflammatory state (especially if caused by diabetes, arthritis, lupus, etc) and causes the body to have difficulty utilizing iron stores in the body. This is not only caused by kidney disease - many chronic conditions, including lupus, rheumatoid arthritis, etc, can cause this. It is called anemia of chronic disease.
Yet another reason people with renal failure become anemic is because the kidneys produce the hormone that stimulates the bone marrow to produce red blood cells - erythropoietin. When the kidneys shut down, they no longer make enough of this hormone, and so the body does not make enough red blood cells - hence anemia.
Another reason is because many people are on dialysis with chronic renal failure, which causes red blood cells to be more likely to break down. Dialysis also causes an inflammatory state in the body (due to interaction of blood with the dialysis machine) and so this increases the anemia of chronic disease and inability to utilize iron stores.