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I am trying to understant this proof on a decomposition property of constructible sheaves on a Noetherian scheme.

But I start to get lost from line 6:"Since F is constructible, there is a finite separable extension..." And also I don't really think the notations make sense: $\kappa(x)\subset\mathop{\mathrm{Spec}}(K)$ does not make sense to me. What is the K? A field? Then the spectrum is just one singleton. So it can not be... Also I don't have a strong clue what they are actually doing. Could someone just explain the main idea of the proof? Thank you!

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