In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that $\mathcal{O}(1)$ is generated by global sections. This is stated at the first of the section that if $\mathbb{P}_k^n = \operatorname{Proj} k[x_0,...,x_n]$ then the $x_0,...,x_n$ give rise to global sections $x_0,...,x_n\in\Gamma(\mathbb{P}_k^n,\mathcal{O}(1))$.
I guess I don't understand this twisted structure sheaf very well, or to be honest I don't think I understand structure sheaves in general as well as I would like. The twisting part seems simple at first- you shift the grading of Proj over and then take the structure sheaf.  Admittedly I don't feel comfortable using the structure sheaf other than using the basic facts about it that Hartshorne gives when it is introduced. If anyone could give some insight as to what's going on here or how I might be able to understand this better it would be much appreciated.
Thanks.

There is a general technique for describing line bundles in topology, differential geometry, complex manifolds,...
We can apply it in algebraic geometry to projective space and it will give you an alternative approach which you might find appealing. Here it is.
(1)
Consider the covering $\mathcal U$ of $\mathbb P^n_k$ consisting of the open subsets $U_i=\lbrace z=[z_0:...:z_n]\in\mathbb P^n_k| z_i\neq 0\rbrace \; (i=0,...,n)$ and the functions $g_{ij}\in \Gamma(U_i\cap U_j,\mathcal O^*_{\mathbb P^n_k})$ defined by  $g_{ij}(z)=\frac{z_j}{z_i}$.
These functions ("a cocycle relative to $\mathcal U$") completely characterize the sheaf $\mathcal O_{\mathbb P^n_k}(1)$ : given an arbitrary open subset $U\subset \mathbb P^n_k$, a section $s\in \Gamma(U,\mathcal O_{\mathbb P^n_k}(1))$
corresponds to  a family of functions $s_i\in \Gamma(U\cap U_i,\mathcal O_{\mathbb P^n_k})$ satisfying $s_i=g_{ij}s_j$ on $U\cap U_i\cap U_j$.
Important example (1) A global section $s\in \Gamma(\mathbb P^n_k,\mathcal O_{\mathbb P^n_k}(1))$ is given by functions $s_i\in  \Gamma(U_i,\mathcal O_{\mathbb P^n_k})$ satisfying $s_i=\frac{z_j}{z_i}s_j$ on $U_i\cap U_j$.
The only possible functions are of the form
$s_i=\frac{L}{z_i}$ where $L=a_0z_0+...+a_nz_n \in (k^{n+1})^*$ is an arbitrary linear form on $k^{n+1}$ and we have thus
$$\Gamma(\mathbb P^n_k,\mathcal O_{\mathbb P^n_k}(1))=(k^{n+1})^*$$
(-1)
Similarly, the functions  $g_{ij}^{-1}(z)=\frac{z_i}{z_j}$ characterize the sheaf $\mathcal O_{\mathbb P^n_k}(-1)$
Important example (-1) A global section $t\in \Gamma(\mathbb P^n_k,\mathcal O_{\mathbb P^n_k}(-1))$ is given by functions $t_i\in  \Gamma(U_i,\mathcal O_{\mathbb P^n_k})$ satisfying $t_i=\frac{z_i}{z_j}t_j$ on $U_i\cap U_j$.
Only $t_i=0$ is possible and we have thus
$$  \Gamma(\mathbb P^n_k,\mathcal O_{\mathbb P^n_k}(-1))=0      $$
An explicit calculation for n=1 (skip if you are fed up with computing!)
In order to really understand what's going on without losing ourselves in indices , let's examine the case $n=1$ and compute $\Gamma(\mathbb P^1_k,\mathcal O(1))$.
A section $s\in \Gamma(\mathbb P^1_k,\mathcal O(1))$ is given by a function $s_0(z)=a+bz+cz^2+...\in  \Gamma(U_0,\mathcal O)=k[z]\; (z=z_1/z_0)$
and a function
$s_1(w)=\alpha +\beta w+\gamma w^2+...\in \Gamma(U_1,\mathcal O) \; (w=z_0/z_1)$
satisfying  (note that $w=1/z$ on $U_0\cap U_1$)
$s_0(z)=a+bz+cz^2+...=z(\alpha +\beta w+\gamma w^2+...)=\alpha z+\beta+\gamma (1/z)+...$ on $U_0\cap U_1$.
This implies that  $a=\beta, b=\alpha$ and that all other coefficients are zero.
Thus, putting $L=az_0+bz_1$, we have indeed proved that $s_0=\frac{L}{z_0}, s_1=\frac{L}{z_1}$ as announced above.
