Let $A$ be a central simple $F$-algebra over a field $F$. A field $E$ such that $F \subset E$ is called a splitting field for $A$ if $A \otimes E$ is isomorphic to $M_n (E)$ (as $E$-algebras) for some $n$.
Why is the algebraic closure of $F$ a splitting field for $A$?
Let $\bar F$ the algebraic closure of $F$ and let $A$ be a csa over $F$.
Then $A\otimes\bar F$ is a central simple algebra over $\bar F$. Wedderburn's theorem tells us that there is a finite dimensional $\bar F$-algebra and an $n\geq1$ such that $A\otimes\bar F\cong M_n(D)$. Since $\bar F$ is algebraically closed, there are no non-trivial finite dimensional division $\bar F$-algebras, so $D=\bar F$.
This means that $\bar F$ splits $A$.