A whole class of schools don't have scholarships at all - including all division 3 schools and the Ivys - so the players at those schools play because they love the sport. These kids are still recruited heavily, though, because these teams still want to win.How does that work exactly? The coach approaches you and asks whether you are willing to go to the school without any scholarship? That somewhat gives away the fact that you are being recruited as a bench player and may not see much playing time. On the other hand, if this helps get you into the dream school you wanted, then it's all good I guess.
Even in division I, no school is allowed more than 9.5 soccer scholarships per team (for men, its more for women) so there are either starters not getting a scholarship or even fewer kids getting full rides. If you look at the stats here: http://www.scholarshipstats.com/soccer.html the average scholarship is often way lower than the tuition. Compare this to football where the top 130 schools can give out 85 scholarships and next 130 can give 65.
Soccer in the US is not an easy path to free college. The only kids getting full scholarships to prestigious programs are at the level where they have to make a serious decision about whether or not to turn pro.