The most basic intuition of weighted projective space is the following:
We define $\mathbb{P}^n_{a_0, \dots, a_n}$ where $a_i \in \mathbb{Z}$ to be the space $\mathbb{C}^{n+1} - \{0\}$ with the relation that $[Z_0: \dots : Z_n] \sim [\lambda^{a_0}Z_0: \dots : \lambda^{a_n} Z_n ]$
I wanted to find a more rigorous definition, like the one given in Harris. However, some of the jargon he uses is slightly unfamiliar to me. He writes that,
Let $a_0, \dots, a_n$ be any positive integers, then we define weighted projective space to be the action of the group $\mathbb{Z}/a_0 \times \cdots \mathbb{Z}/a_n$ on $\mathbb{P}^n$ generated by the automorphisms $$[Z_0: \dots : Z_n] \to [Z_0: \dots : \zeta \cdot Z_i : \dots :Z_n] $$ where $\zeta$ is the $a_i$-th primitive root of unity.
I haven't come across the phrase "group action generated by automorphisms". What is meant by this?