Look at the affine pieces: over the open subset $u \neq 0$, you have a local coordinate $z = u/v$ and your equations can be written as $y = zx$ and $xy = x^6 + y^6$. Substituting $y$ in the second equation gives you $x^2z = x^6 + x^6 z^6$. Now this equation factors as $x^2 = 0$ and $z = x^4(1 + z^6)$; the locus where the first one vanishes is the exceptional divisor, while the second one gives you the closure you are looking for. Non-singularity of the point over $(x, y) = (0,0)$ (which is $(x,z) = (0,0)$) follows from $$ \frac{d}{dz} \big[z - x^4(1 + z^6)\big] \big|_{(x,z) = (0,0)} = 1$$ The other point shows up when considering the other affine piece, $v \neq 0$.

The reason why there are two points over $(x,y) = (0,0)$ is because at that point your curve has two branches. To see that, look at the lowest degree piece of the polynomial defining it (essentially, you are looking here at a neighborhood of the origin in the classical/analytic topology): this is $xy$, which is the union of the two axes. Blowing up pulls these two branches apart.