Proof to an observation of stabilisers and orbits 
Observation:
If $\alpha^{g}=\beta$ then $G_{\beta}=g^{-1}G_{\alpha}g$

Just to get the notation out of the way:
$G_{\beta}= g^{-1}G_{\alpha}g$ is the stabiliser of a point element $\beta$ in a permutation group G of a finite set $\Omega$.
The notation is a bit odd but it is what it is when your professor is an Algebraist.
We need to show $G_{\beta} \subseteq  g^{-1}G_{\alpha}g$ and  $g^{-1}G_{\alpha}g \subseteq G_{\beta}$.
Once we do, we are done with the proof.
I will seek help with the latter. Once I understand the argument I will be able to figure out the former.

Proof:
$g^{-1}: \alpha^{g}=\beta \rightarrow \alpha =  \beta^{g^{-1}}$
$G_{\alpha}:\alpha=\beta^{g^{-1}}:$ fixes $\alpha$
$g:\alpha=\beta^{g^{-1}}\rightarrow \alpha^{g}=\beta$

Indeed, $g^{-1}G_{\alpha}g:\alpha^{g}=\beta\rightarrow \alpha^{g}=\beta$

Hence,
$g^{-1}G_{\alpha}g \subseteq G_{\beta}$

I believe my lack of understanding comes from something more fundamental. To be clear, every line of the argument is understandable but come the conclusion, I am unable to understand how it relates. Say, how is $g^{-1}G_{\alpha}g$ a set and in particular how is it a set of the set $G_{\beta}.$
Would any kind gentleman provide some insights?
 A: I assume $G$ is acting on some set $\Omega$, and for any element $\alpha \in \Omega$, the notation $\alpha^g$ means the image of $\alpha$ under this action, which is also often written as $\alpha \cdot g$.
$G_{\alpha}$ denotes, by definition, the set of elements of $g$ which fix $\alpha$, so $g \in G_{\alpha}$ if and only if $\alpha^g = \alpha$. It is easy to check that $G_{\alpha}$ is a subgroup of $G$. It is called the stabilizer of $\alpha$ in $G$.
For any subgroup $H \leq G$, the set $g^{-1}Hg = \{g^{-1}hg : h \in H\}$ is also a subgroup. Again this is easy to check. So $g^{-1}G_{\alpha}g$ is a subgroup of $G$. The purpose of this problem is to show that $g^{-1}G_{\alpha}g$ is in fact the stabilizer of the element $\alpha^g$.
Restating your argument to make it a bit easier to follow, suppose that $h \in g^{-1}G_{\alpha}g$. Then $h = g^{-1}kg$ for some $k \in G_{\alpha}$, so $k$ has the property that $\alpha^k = \alpha$. Now let us see what happens to $\beta = \alpha^g$ when $h$ acts on it:
$$(\alpha^g)^h = \alpha^{gh} = \alpha^{g(g^{-1}kg)} = \alpha^{kg} = (\alpha^k)^g = \alpha^g$$
This shows that $h$ indeed stabilizes $\alpha^g$. Since $h$ was an arbitrary element of $g^{-1}G_{\alpha}g$, we have shown that
$$g^{-1}G_{\alpha}g \subseteq G_{\beta}$$
Showing the opposite containment is quite similar.
