It is a theorem (due I think to Atiyah) that $\mathcal{F}$ is the classifying space for the topological K-theory functor: $$[X,\mathcal{F}] \cong K(X)$$ for any space $X$. The isomorphism is given as follows: given a map $T \colon X \to \mathcal{F}$, deform $T$ so that $\dim \ker(T(x))$ is constant and assign $T$ to the K-theory class $[\ker T] - [\text{coker}\, T]$. It follows that $\pi_n(\mathcal{F}) \cong K(S^n)$ which can be calculated using Bott periodicity.

EDIT (much later): My answer below is not quite correct, as pointed out by Dan Ramras, because I ignored the importance of base points in the definition of $\pi_n$. See Ramras' answer for the needed corrections.

It is a theorem (due I think to Atiyah) that $\mathcal{F}$ is the classifying space for the topological K-theory functor: $$[X,\mathcal{F}] \cong K(X)$$ for any space $X$. The isomorphism is given as follows: given a map $T \colon X \to \mathcal{F}$, deform $T$ so that $\dim \ker(T(x))$ is constant and assign $T$ to the K-theory class $[\ker T] - [\text{coker}\, T]$. It follows that $\pi_n(\mathcal{F}) \cong K(S^n)$ which can be calculated using Bott periodicity.