Why would anyone go to such lengths to open up their homes to random strangers? Sometimes, people are genuinely interested in meeting new people or showing people around their town. In colleges, prospective students may stay with current students who volunteer to put them up and show them around the school. This is sort of the post-college version of that. Other times, people may have ulterior motives, but the good thing about these sites is that you can write a review of people you meet, host, or stay with, so others can get a better idea of who this person is.
If you don't feel comfortable doing it that way, there's another way you can get free hotel stays. FreeTraveling.com lists mistakes or hotel rooms you can book for free by taking advantage of the TripRewards best rate guarantee. While the city where you need to stay may not be on the list, it is free after all. In turn, you could even make a trip out of going to cities where you can stay for free. As with many things, flexibility in your itinerary will allow you to get the best deals.
Next up: learn the language