The elements of $V/W$ are the sets of the form $x+W$ for $x\in V$, so if $y\in V/W$, then $y$ already is $x+W$ for some $x\in V$. That means that $y=\eta(x)$, so we’ve just proved that $\eta$ is surjective. You still have to prove that $\eta$ is linear and find $\ker\eta$.
To show that $\eta$ is linear, you have to show that for every $x,y\in V$ and scalars $\alpha,\beta$, $$\eta(\alpha x+\beta y)=\alpha\eta(x)+\beta\eta(y)\;.$$ To do that, use the definition of $\eta$ to write down what $\eta(\alpha x+\beta y)$ and $\alpha\eta(x)+\beta\eta(y)$ are, and see if you can see (and explain!) why they must be equal.
The kernel of $\eta$ is the set of vectors $x\in V$ such that $\eta(x)$ is the zero vector in $V/W$. What is that zero vector? Once you figure out what it is, it’s not to hard to figure out which elements $x+W$ of $V/W$ are equal to it.