Finite dimensional central division $\mathbb K$-algebra as a subalgebra of a matrix $\mathbb K$-algebra

Write $d=\dim_K(D)$. Consider the set $D^m$ for a natural number $m$. Then $D^m$ is a right $D$-module, and has dimension $md$ as a vector space over $D$. Choose a basis $B$ for $D^m$ over $K$. Each right multiplication map $\rho_a:x\to xa$ on $D^m$ is a $K$-vector endomorphism of $D^m$. Expressing it in terms of the basis $B$ gives a matrix $N_a$. Then $a\to N_a$ is a homomorphism from $D$ to $M_{md}(K)$. As $D$ is simple, its image is isomorphic to $D$ and is a subring of $M_{md}(K)$.

Conversely consider an embedding $\phi:D\to M_n(K)$. Then $K^n$ becomes a right $D$-module via $v\cdot a=v\phi(a)$. For a division ring, module theory follows the same lines as vector space theory over a field: every module is free. Thus $K^n\cong D^m$ as a $D$-module for some $m$. Then $n=md$ on counting dimensions over $K$.

If $D$ is a unital subring of $M_r(K)$ then $K^r$ is a unital $D$-module. But $D$ is a division ring so this means $K^r$ is a $D$-vector spaces, i.e. $K^r=D^m$ for $m=\dim_D K^r$. Evidently $r=m\dim D$.