I was wondering how one computes $\mathcal{O}_X(U)$ for $U=X\setminus \mathbb{V}(x_0^2+x_1^2+x_2^2)$ and $X=\mathbb{P}^2$ by considering the usual cover of $\mathbb{P}^n$ by affine charts $U_i$. Apparently this can be done by first computing $\mathcal{O}_X(U\cap U_i)$ and then considering the restriction of functions to the overlaps $U\cap U_i \cap U_j$.
I understand the definition of $\mathcal{O}_X(U)$ in terms of it being the ring of regular functions $F:U\rightarrow k$ and the idea that one can glue regular functions that agree on intersections of open sets but am not sure how to bring this together to properly calculate this.
So first of all, $\mathbb{A}^2 \setminus V(x_{0/2}^2 + x_{1/2}^2 + 1)$ is an open subvariety, not a closed subvariety, of $\mathbb{A}^2$. Therefore, you can't write its ring of regular functions as $k[x_{0/2}, x_{1/2}]/I$ for some ideal $I$. However, the second thing you wrote is correct: you want to consider the localization of $k[x_{0/2}, x_{1/2}]$ at the function $x_{0/2}^2 + x_{1/2}^2 + 1$, which is indeed given by $k[x_{0/2}, x_{1/2}, z]/(z(x_{0/2}^2 + x_{1/2}^2 + 1) - 1)$. The localization basically means that you're allowed to invert $x_{0/2}^2 + x_{1/2}^2 + 1$.
Now you take the intersections of the coordinate rings on the three different charts. Why is it enough to simply take the intersection of the coordinate rings? Because the ring of regular functions of $\mathbb{P}^2 \setminus V(x_0^2 + x_1^2 + x_2^2)$ is necessarily a subring of $k(x_0, x_1, x_2)$, which is the field generated over $k$ by all rational functions in $x_0, x_1, x_2$. The intersection of the three coordinate rings is given by rational functions of the form $\frac{g(x_0, x_1, x_2)}{(x_0^2 + x_1^2 + x_2^2)^m}$, where $g(x_0, x_1, x_2)$ is a polynomial of degree $2m$. [Do you see how to prove this last bit?]