From the [long exact sequence of homotopy groups][1] associated to the fibration $U\to EU\to BU$, one has that $\pi_{2k+1}(BU(m))=\pi_{2k}(U(m))$ (since $EU$ is contractible, and for $k>0$). From the proof of the [Bott periodicity theorem][2], $\pi_k(U(m))$ stabilizes for $m$ large (and equals $\pi_k(U)$). Moreover, $\pi_k(U)=\pi_{k+2}(U)$. So for even $k$, $\pi_k(U)=\pi_0(U)=\mathbb{Z}$. So for $m$ large, $\pi_{2k+1}(BU(m)) = \mathbb{Z}$. 


  [1]: https://en.wikipedia.org/wiki/Homotopy_group#Long_exact_sequence_of_a_fibration
  [2]: https://en.wikipedia.org/wiki/Bott_periodicity_theorem