The most concrete way to express the relation between Dirac matrices and homotopy groups of classical groups is given by the following construction.

Let $(\gamma^\mu)_{\mu=1, \dots ,d}$ be $n \times n$ Dirac matrices (in Euclidean signature), i.e. satisfying

$\{ \gamma^\mu , \gamma^\nu \} = 2 \delta^{\mu \nu}$

Then the map

$\mathbb{R}^d \rightarrow M_n (\mathbb{C})$

$(x_\mu)_{\mu=1, \dots, d} \mapsto \sum_{\mu =1}^d \gamma^\mu x_\mu$

gives in restriction to the unit sphere $\sum_{\mu=1}^d (x_\mu)^2=1$ a map

$\mathbb{S}^{d-1} \rightarrow GL_n (\mathbb{C})$

(indeed, in this case: $(\sum_\mu \gamma^\mu x_\mu)(\sum_\nu \gamma^\nu x_\nu)= \sum_{\mu=1}^d (x_\mu)^2=1_n$).

Taking the homotopy class of this map, we get an element in $\pi_{d-1}(GL_n (\mathbb{C}))=\pi_{d-1}(U_n)$.

If the Dirac matrices are real, we get a map

$\mathbb{S}^{d-1} \rightarrow GL_n (\mathbb{R})$

and so an element in $\pi_{d-1}(GL_n (\mathbb{R}))=\pi_{d-1}(O_n)$.

We get a map from possible Dirac matrices to homotopy groups of unitary groups (or orthogonal groups). Up to appropriate equivalence on both sides, this is a one to one correspondence : see

http://www.maths.ed.ac.uk/~aar/papers/abs.pdf

for the precise (and original) statement. In particular, under this correspondence, Bott periodicity on the Dirac matrices side matches with Bott periodicity on the homotopy groups side.

For a physical interpretation in the context of D-branes, see https://arxiv.org/abs/hep-th/9810188