First of all: $\rho$ is usually used for a Galois representation; in the context, I'm pretty sure it's supposed to be the Galois representation given by the Tate module of an elliptic curve over $\mathbf{Q}$, which is a continuous homomorphism $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to \operatorname{GL}_2(\mathbf{Z}_p)$ for some prime $p$. $\operatorname{Sym}^2 \rho$ is the symmetric square Galois representation into $\operatorname{GL}_3(\mathbf{Z}_p)$.
In the $H^1$ bit: $\mathbf{Q}_\Sigma$ is the maximal extension of $\mathbf{Q}$ unramified outside a finite set $\Sigma$ of primes, and $H^1(\mathbf{Q}_\Sigma / \mathbf{Q}, -)$ is shorthand for the group cohomology $H^1(\operatorname{Gal}(\mathbf{Q}_\Sigma / \mathbf{Q}), -)$ (more strictly, continuous gropu cohomology, which respects the Krull topology on the Galois group). We can plug in for the "$-$" any Galois representation factoring through the canonical map
$$\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to \operatorname{Gal}(\mathbf{Q}_\Sigma / \mathbf{Q}),$$
i.e. any Galois representation unramified outside $\Sigma$; the Galois representation attached to an elliptic curve (or its symmetric square) satisfies this as long as $\Sigma$ contains $p$ and all primes dividing the discriminant of the elliptic curve.
As for $\mathbf{Q}_p / \mathbf{Z}_p$: this is the "Prufer group", an infinite torsion group isomorphic to the direct limit of cyclic groups of order $p^n$ over all $n$. It's useful here because taking homs into $\mathbf{Q}_p / \mathbf{Z}_p$ gives a well-behaved duality theory for topological modules (one form of Pontryagin duality).
The only thing I haven't explained yet is the subscript $f$ in $H^1_f(...)$. This is perhaps the most delicate thing here: it's the "finite part" of the Galois cohomology group $H^1(...)$, a certain canonical submodule defined in a famous 1990 paper of Bloch and Kato. $H^1_f(\mathbf{Q}_\Sigma / \mathbf{Q}, \rho)$ is closely related to the Selmer group of the elliptic curve, so the $H^1_f$ functor is a sort of "Selmer group of an arbitrary Galois representation". The size of the Bloch--Kato Selmer group of $\operatorname{Sym}^2 \rho$ is important here because it determines how the deformation theory of $\rho$ behaves.
A good place to learn more about these things would be the book by Cornell, Silverman and Stevens, which user33240 has already linked to.
I will try to give you a VERY quick overview on the strategy of the proof of FLT. Of course I cannot avoid to use very technical tools, such Galois representation. If you don't know about these stuff, I hope you can at least follow the "shape" of the argument.
The starting observation is the following: for $n\in\mathbb N$, let FLT($n$) be the statement "there are no triples $(a,b,c)$ of integers with $abc\neq 0$ such that $a^n+b^n=c^n$". Then it is elementary to see that FLT($d$) implies FLT($n$) whenever $d\mid n$. Therefore it is sufficient to prove FLT($p$) for every odd prime $p$ and for $n=4$ in order to prove FLT($n$) for all positive integers $n\geq 3$. The cases $p=3$ and $n=4$ were already known to Euler, I believe, and can be proven by elementary methods, so we can assume that $p\geq 5$.
Now the non-elementary math comes in. Suppose that there is a triple of integers $(a,b,c)$ which contradicts FLT($p$) for some prime $p$. One can construct the following elliptic curve over $\mathbb Q$:
$$E_{a,b,c}\colon y^2=x(x-a^p)(x-b^p)$$
It is possible to show (assuming wlog that $(a,b,c)$ is a coprime triple, $a\equiv -1\bmod 4$ and $2\mid b$) that this is a semistable elliptic curve whose conductor is $\prod_{l\mid abc}l$.
Moreover, Serre and Frey proved the following theorem: let
$$\overline{\rho}_{a,b,c}\colon \text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to GL_2(\mathbb F_p)$$
be the residual Galois representation at $p$ attached to $E_{a,b,c}$. Then:
- $\overline{\rho}_{a,b,c}$ is absolutely irreducible;
- $\overline{\rho}_{a,b,c}$ is odd;
- $\overline{\rho}_{a,b,c}$ is unramified outside $2p$ and flat at $p$
The key idea is then the following: suppose that we can show that for every (semistable) elliptic curve $E$ over $\mathbb Q$ the representation $\overline{\rho}_E$ is the residual representation of the $p$-adic Galois representation attached to a weight $2$ newform $f_E$. Then we can apply a theorem of Ribet, using the properties of $\overline{\rho}_{a,b,c}$, to show that for $\overline{\rho}_{a,b,c}$ we can choose such a newform in $S_2(\Gamma_0(2))$. But then we have a big problem: this space is $0$-dimensional! Therefore we are led to a contradiction, and there cannot be any triple $(a,b,c)$ contradicting FLT($p$).
This whole argument was already known long before Wiles' proof. But the thing that was missing was the proof of the (nowaday) so-called modularity theorem: every elliptic curve over $\mathbb Q$ is modular, i.e. its $p$-adic Galois representation is the $p$-adic Galois representation attached to a weight $2$ newform.
Even though this was the only missing part, it is by far the most difficult one! Wiles was able to prove it for semistable curves, which was enough for proving FLT, but some years later the result has been improved to all elliptic curves over $\mathbb Q$ by Breuil, Conrad, Diamon and Taylor.
Anyway, if you are interested in more details, you can read the (amazing) first chapter of "Modular forms and Fermat's last theorem", by Cornell, Silverman and Stevens eds. Of course it is a math textbook, so you need some background in these type of topics in order to understand it.
Best Answer
This is a nice idea but the quote is about the equation $$x^n+y^n=\color{red}1$$So, assume that $$A^3+B^3=C^3$$We get $$\left(\frac AC\right)^3+\left(\frac BC\right)^3=1$$
And also indeed $$(2A)^3+(2B)^3=(2C)^3$$ And we again get$$\left(\frac {2A}{2C}\right)^3+\left(\frac {2B}{2C}\right)^3=\left(\frac AC\right)^3+\left(\frac BC\right)^3=1$$ Which is the same solution as before.