References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ I have read (Theorem 29.3, line 7, pg 144) the following statement: 

$\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$.  

and 

Therefore $\pi_i(St_k(U(n))=0$ for $i \le 2(n-k)$. 


What results are used here and where can I find a reference of this? 

I think they have made an off-by-$1$ error, they might have been thinking about the map $BU(n-k) \to BU(n)$. (The notes you're reading are unpublished and there is a disclaimer at the start about not being proofread, so an off-by-$1$ error isn't so bad.)
A great reference for a lot of results like this is Mimura and Toda's "Topology of Lie Groups " I and II. They compute a lot of homotopy and homology groups for classical Lie groups and their classifying spaces. For this particular example, look at Corollary 3.17 on page 68. The statement includes: 

... For $i < 2n$, $\pi_i U(n) \cong \pi_i U(n+1)$, ...

Changing variables we see $\pi_iU(n-1)\cong \pi_i U(n)$ for $i < 2(n-1)$, and their argument actually also shows surjectivity in degree $2(n-1)$.
Typically you use the long exact sequence of the fibration
$$ U(n - 1) \to U(n) \to S^{2n - 1}$$
to compute the connectivity of $U(n - 1)\to U(n)$. The degree $i = 2n - 2$ is the last degree where $\pi_i S^{2n-1} = 0$ so our map induces surjective homomorphisms in all degrees $i \leq 2(n-1)$, and moreover they are injective if $i < 2(n-1)$. The (correct) connectivity of your map, and hence the Stiefel manifold, follows.
