Why create a subkey in the digital signature algorithm In the signing step, wikipedia, a subkey is created.

A key k is randomly chosen from 1..q-1.


What's the use of that k? 

If the k is to make less exposure to the private x; however, x is inevitably used in the signing part when computing s.
What's the use of that k? 


The following lines of the wiki tell you:

Compute $r := ( g^k \bmod p  )  \bmod    q$

Compute $s := ( k^{−1} ( H(m) + xr  ) ) \bmod    q$

That is, it is used to compute both $r$ and $s$.

However, that's not what your really asking.  You're asking "why did they design DSA that way?"

Well, it's actually a variation on a noninteractive Schnorr proof of knowledge of a discrete log (in particular, of the private key); the modifications are there to reduce the size of the signature.

And, since Schnorr requires a "commitment" (what you call a subkey), well, so does DSA