Understanding the "gap theorem" in geometric spanners
Let's write $C = (1+2/w) wt(MST(S))$. The set $E$ satisfies the $w$-gap property and has $m$ edges. It can be partitioned into 2 sets: 

One set is $E'$, which is a special subset of $E$. It is chosen such that we can easily prove that its weight is at most $C$. 

The other set $E''$ (which is $E \setminus E'$) has $m/2$ edges; we only know that it satisfies the $w$-gap property. By the induction hypothesis, the weight of $E''$ is at most $C \log(m/2)$. 

From this, it follows that the set $E$ has weight at most $C + C \log(m/2) = C \log m$.