The subsidized government loans artificially lowers the cost of college, which has the effect of increasing BOTH supply and demand for college degrees. Trading out subsidized loans for private loans will put the cost of college back at its economically real price and decreasing number of students and demand for degrees.
However, I don't think this economics 101 description is sufficient to describe what is really happening. Supply of college degrees has been artificially increased in a sort of self-replicating process. The government triggered it by deciding that all people need to have access to college if they choose, and they instituted a combination of policies (Pell grant, subsidized loans, etc.) to make this happen. Well a lot of extra people did indeed go get that degree to get higher wages, and employers found that they could more easily demand a college degree from applicants. And also this degree would not entail paying them as much as it used to, because the degree is not as rare as it used to be. As more employers demanded degrees, more people had to go to school to get a degree. Here is where the key thing happens. These people have been ABLE to get the degree because they have a guaranteed student loan from the government, and so even more college degrees flood the market and the cycle repeats.
If it had been a private student loan system, and that private system was working responsibly, then they would have drastically slowed down providing student loans a long time ago, and the college degree numbers would have corrected. If a private loan system was not working responsibly then you get basically the same effect as the housing bubble.