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I'll assume you understand everything down to the point where the error is bounded by 2 / 5! * | x^5|, and that the fourth-degree polynomial is 1 + x + x^2 / 2 + x^3 / 3! + x^4 / 4!. If you have specific questions on how any of these results were obtained, please ask. I'll be glad to add to the book's explanation.

If -.5 < x < .5, then | x^5 | will take its max value at either x = -.5 or x = .5, at both of which | x^5 | = .5^5 = (1/2)^5 = 1/32.

So the max possible value of 2 / 5! * | x^5| is 2 / 120 * (1/32) = 2 / (120 * 32) = 1 / (60 * 32), which is close to and a bit less than .0006.

So .0006 is the error bound.