How exactly do altcoins gain value? There are so many altcoins out there and honestly you can see connection among these in terms of the drop and rise of price.
if you want to understand altcoin market then search for "Penny Stocks", that should give you a pretty good idea what you are dealing with. and in this market you have the addition of manipulation in form of pump and dump and lots of newbies and underage kids who got in because the market was open unlike other markets which you can't get in with 20 bucks!
so mainly the altcoins are gaining value based on speculation and thin air.
your question in the title is asking a very different thing though. it implies altcoins are fork of something which is true in 90% of them. they fork from other projects like bitcoin most of the time.