Your question appears to have two parts:
- How to comply with an LGPL backed source request.
- Why the authors of a library you included would request their own source.
Source distribution mechanics
The first question is pretty mechanical and fairly straightforward. Namely: tar / zip up the files that were used and send them to the requestor. It makes no difference who the requesting person is. You provide the source, as requested.
If you were providing the source via FTP, you could verify the FTP repository was working and have them retrieve the source from there. It's possible that not all variations of the GPL1 licenses will support that approach.
The safest version for distribution or conveyance is directly sending the source.
Rationale of request from library author
Part of providing Free software (that's "Free as in Freedom" means following up and making sure that downstream consumers of the Free software are also complying with the terms of the license.
It's one thing to put up an FTP link or provide a disclaimer of "source available upon request". But it's another level to actually verify that the FTP links do provide the source or that the source is actually available when requested.
It sounds like the creator of the library you used wanted to verify that you were complying with the terms of the license. They (obviously) didn't need their own source code back. They may have been concerned that you made modifications without re-releasing them, too. Given the size of VLCkit, I don't think that was the case. The most likely answer then is they want to make sure you're complying with the terms of the *GPL1 licensing that was used.
And based upon your follow-up comment:
I asked and he replied. Just zip up the source that I used and send it to him. No problem.
It sounds like they were making sure that you were doing your part in the Free software movement.
1 I'm writing under the presumption that other packages were used that were also either LGPL, AGPL, or GPL licensed.