If you are wondering about the inner mechanisms of the chain rule for partial derivatives, you can read about it on Paul's Online Notes, specifically under 'Case 2'. If you think of $\nabla f(x,y)$ as $\frac{\partial f}{\partial x}\boldsymbol{i} + \frac{\partial f}{\partial y}\boldsymbol{j}$, you know that at point $(2,1)$ $\frac{\partial f}{\partial x} = 4$ and $\frac{\partial f}{\partial y} = 3$.
Next you can set $u(x,y) = x^2 + y^2$ and $v(x,y) = xy$ and use the partial derivatives $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, $\frac{\partial v}{\partial x}$, and $\frac{\partial v}{\partial y}$ to create a new expression...
$\nabla f(u,v) = (\frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x})\boldsymbol{i} + (\frac{\partial f}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial y})\boldsymbol{j}$
From the logic of the first paragraph, you know what the values of $\frac{\partial f}{\partial u}$ and $\frac{\partial f}{\partial v}$ are at the point $(2,1)$. The expressions in the parentheses transform that $f(u,v)$ result in terms of the $x$ and $y$ used in $G(x,y)$.