The important point is that $\Gamma_w$ is a HNN extension, i.e., an extension of $\Gamma$ (or rather of $\Gamma$ already freely extended by $s$) by a new element $t$ and using an isomorphism $\alpha\colon H\to K$ between two subgroups of $\Gamma$ two subgroups $H,K$ of $\Gamma$ in such a way that conjugation with $t$ applied to $H$ acts like $\alpha$; in other words, $\langle\, S\mid R\,\rangle$ becomes $\langle\,S,t\mid R,\forall h\in H\colon t^{-1}ht=\alpha(h)\,\rangle$. We clearly have a homomorphism $\langle\, S\mid R\,\rangle\to \langle\,S,t\mid R,\forall h\in H\colon t^{-1}ht=\alpha(h)\,\rangle$ by sending $S$ to itself. A priori, this need not be an injective homomoprhism, and it is not obvious that it should be, but it is the key result of the theory of HNN that it is indeed injective.
The subgroups $H,K$ we use here are two free subgroups on $n$ generators contained in $\langle \Gamma,s\rangle$, one generatied by $s^ia_is^{-i}$, $1\le i\le n$, and free because none of the $a_i$ is $1$, the other generated by $s^iws^{-i}$, $1\le i\le n$, and free because $w\ne 1$. The isomorphism $\alpha$ is just that sending $s^ia_is^{-i}$ to $s^iw^{-1}s^{-i}$ (the inverse because of the way the relation is written). As $H,K$ are free, it suffices to add the HNN relations for the generators only.
In total we have thus two injections
$$ \Gamma = \langle\,A\mid R\,\rangle\hookrightarrow\langle\,A,s\mid R\,\rangle\stackrel{\text{HNN}}\hookrightarrow\langle\,A,s,t\mid R,\forall i=1,\ldots, n\colon s^ia_is^{-i}=s^iw^{-1}s^{-i}\,\rangle=\Gamma_w$$
with
$$ H=\langle\,sa_1s^{-1},\ldots, s^na_ns^{-n}\,\rangle \cong F_n\cong \langle\,sws^{-1},\ldots, s^nws^{-n}\,\rangle=K.$$
For more examples see:
C. Campbell, E. Robertson,
A deficiency zero presentation for $SL(2,p)$.
Bull. London Math. Soc. 12 (1980), no. 1, 17–20.
and, more recently,
C. Campbell, G. Havas, C. Ramsay, E. Robertson, Nice efficient presentations for all small simple groups and their covers. LMS J. Comput. Math. 7 (2004), 266–283.
C. Campbell, G. Havas, C. Ramsay, E. Robertson,
All simple groups with order from 1 million to 5 million are efficient. Int. J. Group Theory 3 (2014), no. 1, 17–30.
Here a superperfect group (a perfect group with trivial Schur multiplier) is efficient if and only if it admits a balanced presentation.
The last two papers cover all groups of order up to 5 million.
Best Answer
One can prove quite simply a much more general result due to McKinsey, viz.
