As Luis Casillas said in his comment, the spec of Salsa is short and well explained. Namely, if you are interested in the expansion function, you can find a description with two examples (one with a 32-byte key, and one with a 16-byte key).
Short answer: when dealing with a 16-byte key, you only have to change a constant and append the key to itself.
More details bellow.
Salsa keystream generator, noted $Salsa20_k(n)$ for a key $k$ and a nonce $n$, is based on a hash function : $hSalsa20$ (called $Salsa20$ in the spec). This hash function works as follow :
if $x$ is a 64-byte sequence, $hSalsa20(x)$ is a 64-byte sequence.
Now, let's see the role of the key and how it is managed. Whether you're using a 32-byte key or a 16-byte key, there is no real key expansion as you can find in Blowfish. Instead, the key $k$ is divided in two 16-bytes subkeys $k_0$ and $k_1$ (if $k$ is 16 bytes long, $k_1 = k_0$). Then, a 16-byte constant is added to the state, and that's where there is a difference : the value is not the same depending on the key size:
- for a 16-byte key, this constant is "expand 16-byte k"
- for a 32-byte key, this constant is "expand 32-byte k"
This constant is divided into 4 equal part, let's call them $\sigma_0,\sigma_1,\sigma_2$ and $\sigma_3$, 4-byte each.
The keystream is produced as follow :
$$Salsa20_k(n) = hSalsa20(\sigma_0\ ||\ k_0\ ||\ \sigma_1\ ||\ n\ ||\ \sigma_2\ ||\ k_1 || \sigma_3)$$
where $||$ denotes the concatenation.
Notice that in both case, the argument of $hSalsa20$ is 64-byte, so 64-byte of keystream is produced. You only have to increment the nonce, and start over to generate more keystream.
