References:

Artin, M.
*Brauer-Severi varieties*. Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), pp. 194–210,
Lecture Notes in Math., 917, Springer, Berlin-New York, 1982.

Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre
**The Book of Involutions**. (English summary)
With a preface in French by J. Tits. American Mathematical Society Colloquium Publications, 44. American Mathematical Society, Providence, RI, 1998

Let $A$ be a central simple $K$-algebra of degree $n$ (i.e., of rank $n^2$). Let $Gr(n, A)$ be the subspace of the grassmannian of rank $n$ dimensional subspaces. Let $X \subset Gr(n, A)$ be the subvariety of subspaces which are also right ideals of $A$. Then $X$ is the corresponding space, called the *Severi-Brauer*, *Brauer-Severi*, or *Chatelet* variety of $A$, and is denoted here by $SB(A)$.

When $A = End_K(V)$ is trivial, every right ideal of $A$ of dimension n (as a $K$-vector space) is of the form $\operatorname{Hom}_K(V, U)$, where $U$ is a 1-dimensional vector space. So the space is $\mathbb{P}_K(V)$.

I am not sure of an explanation in the other direction as nice. It is the case that for a Severi-Brauer variety $SB(A)$, you can consider the following subbundle $\mathcal{I}$ of the constant bundle $SB(A) \times A$. The bundle $\mathcal{I}$ consists of pairs $(I, a)$, where $a \in I$. Then $\mathcal{I}$ is a locally free of rank $n$ and the global endomorphism ring $\operatorname{End}_{SB(A)}(\mathcal{I})$ is isomorphic to $A$ (or the opposite algebra of $A$, I can't remember which).

However, this doesn't answer your question of how to start with a twist of $\mathbb{P}^{n-1}$ and nicely produce a central simple algebra of rank $n^2$. I think for that you may have to argue by galois descent.