$$\mathbb{P}\left(\bigoplus_{i=0}^rH^0(\mathscr{O}_{\mathbb{P}^1}(d))\right)$$
From the context of the book, I think it is more or less the space of all $r$-tuplets of homogeneous polynomials of two variable of degree $d$.
But can someone explain this notation bit by bit? First, what does $\mathbb{P}$ outside represent? Then is $H^0$ the zero-th cohomology? But I only know sheaf cohomology which is usually like this: $H^i(X,\mathscr{F})$, the space and the sheaf. And last how to understand this sheaf $\mathscr{O}_{\mathbb{P}^1}(d)$?
For any vector space $V$ over a field $k$, $\mathbb P(V)$ denotes the projective space defined by $V$, which is the space of nonzero elements of $V$ modulo the relation $v\sim w$ if $v=\lambda w$ for some $\lambda\in k^\times$. Note that the usual projective space $\mathbb P^n$ is just $\mathbb P(k^{n+1})$ in this notation.
$H^0$ is indeed the zeroth sheaf cohomology, which is simply the vector space of global sections of the sheaf. You are right that technically we should specify what space the sheaf is considered over, but here it should be clear that the sheaf $\mathcal O_{\mathbb P^1}(d)$ is considered over $\mathbb P^1$.
Finally, $\mathcal O_{\mathbb P^1}(d)$ is a special line bundle over $\mathbb P^1$. It can be thought of as the sheaf of all degree $d$ homogeneous rational functions on $\mathbb P^1$ (or degree $-d$, the convention depends on the author). You can read up the precise definitions for instance here.