I do not understand the purpose of the end of the last but one line and the last line in the following snippet from Jech's book: if $2^{\operatorname{cf}\kappa}\geq \kappa$ then $\kappa^{\operatorname{cf}\kappa}=2^{\operatorname{cf}\kappa}.$ Namely, why the first condition with $\geq$ is assumed ?
I'll turn my comments into an answer to get this off the unanswered list...
The results of all cardinal exponentiation calculations are determined from the $\gimel$ function ($\kappa\mapsto \kappa^{\operatorname{cf}\kappa}$) on singular cardinals, and the continuum function ($\kappa\mapsto 2^\kappa$) on regular cardinals. Jech demonstrates this in corollary 5.21, before the section on SCH. (Note that the gimel function and the continuum function give the same thing on regular cardinals, so we can really say that cardinal exponentiation is determined by the gimel function alone, which is how Jech puts it.)
What Jech is getting at in this confusingly terse passage is that SCH allows us to reduce the gimel function on singular cardinals to the continuum function on regular cardinals. He splits it up into two cases: The first case is: $2^{\operatorname{cf}\kappa}\ge \kappa,$ then (in ZFC alone), we have $\kappa^{\operatorname{cf}\kappa}= 2^{\operatorname{cf}\kappa},$ which is the value of the continuum function on the regular cardinal $\operatorname{cf}\kappa.$ The second case is $2^{\operatorname{cf}\kappa}< \kappa,$ in which the SCH says $\kappa^{\operatorname{cf}\kappa}$ takes its minimum possible value $\kappa^{\operatorname{cf}\kappa}=\kappa^+.$
So, since we can get gimel in terms of values of the continuum function on regular cardinals we should be able to reduce any computation $\kappa^\lambda$ to that as well. Jech puts this together in theorem 5.22, just after the passage you quoted.