I think you don't quite understand how RSA signatures work (and why they are the size they are).
When generating an RSA signature, we follow a two-step process:
We take that hash of the message we're signing, and convert (and pad) it into an integer $M$ which is between 0 and $N$ (where $N$ is a large integer that specified by the RSA key)
We use the RSA private key, and convert $M$ into a signature $S$, which is also an integer betwen 0 and $N$
The resulting integer value $S$ is the signature.
From this description, it should be obvious that $S$ is essentially the same size as $N$. So, how big is $N$?
Well, for the RSA keys that you use, it is 2048 bits long (which, in the context of RSA, means that it is an integer between $2^{2047}$ and $2^{2048}-1$).
Hence, the RSA signatures you see are 2048 bits long, not because someone decided to pad out the hash an extra 1888 bits, but instead when they generated the RSA key, they decided to make that key 2048 bits long. They didn't add 1888 bits to the hash because they thought 1888 was a nice number; instead, they extended the hash to 2048 bits because whoever generated the RSA key thought 2048 was a nice number.
Now, you might ask: why did they pick a key of that size? Well, more than anything else, because that's a round number. You can generate RSA keys of just about any size (however, too small keys can be broken, and too large keys just waste time), and practically speaking, there's little difference between an RSA key of 2048 bits and one of 2000 bits. However, 2048 bits is what everyone expects nowadays, and so that's what everyone uses.
You asked for a reference about how the RSA padding process works; the one that immediately comes to mind is PKCS #1, which explains exactly how to generate RSA signatures (using several different padding methods).
