What does it mean to co-sign a letter of recommendation?
Co-signing something simply means that the co-signer also approves of the letter. For example if I wrote:

To whomever it may concern,
Seankala is a brilliant student.
Sincerely, Allure

If someone else were to co-sign it, it would become:

To whomever it may concern,
Seankala is a brilliant student.
Sincerely, Allure &amp; [someone]

It means that that someone also thinks Seankala is a brilliant student.
Co-signing something doesn't mean one wrote or even helped write it. Here's an example of a co-signed petition. The petition itself was obviously not written by 12,000 people, but you can be sure they all approve of it.