I'm confused with the yellow parts, what is the clearly meaning of recurrence, explain it clearly please

and why we but the equal sign instead of greater than or equal, Also where is ( -c n + n ) in the last step, should I give c value or what? what is the meaning of the last line ! (hold as long as c>=1 )

T (n) = 2T ([n/2]) +n, We guess that the solution is T(n) = O(n lg 1). Our method is to prove that T(n) 0. We start by assuming that this bound holds for (n/2), that is, that T([n/2]) 1