Interesting question. I've known about faucets for a while and now that Bitcoin is rising, I'm a little upset that I haven't been claiming up until now so I've started again. As such, it's never really occurred to me to doubt them, but I can see why you might be hesitant.

Firstly, who's giving away Bitcoins?
Well, anyone could do it. You could, too. So, just people really. You could build your own faucet and give away satoshi.

Why would you do it?
Primarily the advertisement money. A lot of faucets insist that you disable ad-block on their sites because they're ad-based. There are other revenue streams for faucet owners, though. For example, you'd place links on your website to other faucets. Links that look like this: http://btcclicks.com/?r=c50596c5. The "r" part in this probably stands for "referrer" and the following number is my unique code so if you sign up here through that link, I'll profit, too. All faucets are based around this mechanic, even Coinbase, the wallet I linked has a referral scheme. So when people come to your faucet for BTC, they'll leave through your links for even more, and you'd profit from that.

Taking this logic one step further, you could do things that people aren't even doing right now. Such as advertising your own personal content. A popular faucet is a website with constant daily traffic, so you could get a lot views for YouTube videos and you can also sell that space to a funny, new YouTuber or blogger.

But you know, Bitcoin used to be worth a lot less than it is now. It used to be worth nothing at one point, so websites like this were really profitable back then as they would make a lot more from the ads than they would lose from giving away satoshi. Now, faucets reduce the amount of bitcoin that they offer so that they still profit.

But it's more than just that.
Receiving BTC for free is a part of Bitcoin culture. Always has been and I hope it always is. In fact, I shared one of my finds earlier this year. If you embed a snippet of HTML into your website that graphically displays the current price of one bitcoin, you'll receive a little BTC for free.

As for where the money comes from, well...
If I read Freebitco.in's about page correctly, they mine their own. They run an operation where they generate the BTC and give it away. They don't run adverts, so I'm curious about them. They do give you the option to gamble, so they probably keep a lot of the BTC they give out. They've been established for a long time, too, so they're hard for me not to trust. I mean, people drain a lot of faucets so if you stop paying them, they will get mad and they will tell others.

Unfortunately, my knowledge on the topic is limited. I'm sure there must be other ways to get your hands on the BTC to gift out - other than just buying it, of course - and there must be more to the profit side of things but I only know a little bit from what I've learned here and there.

If you want a more comprehensive write-up, here's a good article on the topic. It's better than anything I could ever write.

Also, if you don't know, they don't give out whole bitcoins. Only fragments of one. If you want to find out how many satoshis go into one BTC, you can use a calculator.