This is the top Apple Support hit for "Apple Hardware Test El Capitan", so although the post is from last year still worth adding to for anyone else who comes hunting like I did.
This may be obvious to some, but not to all readers: the USB stick has to be formatted correctly.
Basically, your USB stick needs to formatted as HFS+, otherwise known as Mac OS Extended, with a GUID partition table scheme.
Almost all USB sticks you might purchase are formatted to allow them to be read from and written to by both Mac and Windows computers, but not in such a way that they can be used as a bootable disk for a mac.
So you could happily copy across the relevant AHT files, but you still won't be able to boot from your stick if the stick is the wrong format.
There's a good step-by-step on how to format your stick correctly, which does not assume any technical knowledge, here: http://www.macworld.com/article/2055589/storage/how-to-format-a-startup-drive-fo r-a-mac.html
If you format your stick as per the article above, then follow the instructions on https://github.com/upekkha/AppleHardwareTest to download and install the AHT on your stick, it should work. Note: should...
I just did exactly that on a 2007 Macbook Pro 3,1 running El Capitan, and the AHT is chugging away on it next to me right how.