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What is “asymptomatic microscopic hematuria” and what causes this minute amount of blood to show up in a urinalysis without any symptoms to the patient?

As your question notes, microscopic hematuria refers to the presence of red blood cells in the urine in such small numbers that they are detectable only by microscopic examination of the urine sediment. This is in contrast to gross hematuria in which the urine is red due to blood. The term asymptomatic indicates that the finding is not associated with pain or other symptoms.
The answer to the second part of your question is much more complex. The fact that blood is not grossly visible does not exclude a significant or potentially serious cause for the problem. This could include infection, inflammation of the kidney (nephritis), stones or tumors in the kidney or bladder, and cysts of the kidneys. In many instances, a urologic evaluation fails to establish such a condition and the problem is considered to be the result of “benign hematuria” for which no treatment is needed. This problem may be detected in childhood and on a familial (inherited) basis.