The first thing to know about this question is that it is about a hotly-debated political topic. People on different ends of the political spectrum have different answers to offer. There is NO answer that everyone agrees is true or most meaningful. To see the opposing arguments and weigh them, which a good essay answer requires, you have to Google the topic and read both sides of the debate.
In the United States and even in Great Britain, Federal and State governments provide tuition relief at Government colleges in two ways: (1) Government support for colleges' budgets reduces the full price that students have to pay to attend those public colleges for all students who attend them, and (2) government "need-based" support for low-income students makes it possible for students to attend college who otherwise would not have been able to pay even the state-subsidized tuition.
So Government provision of university education reduces the tuition both for the wealthy and even more so for the poor.
Governments also offer loans, grants and scholarships that help low-income people pay the tuition at public universities.
"Supply-side" right-wing conservative economists offer arguments like this one:
"With tuition money easy to obtain through loans, demand for a college education increased. With the increased demand came higher tuition costs. This easy money is the primary reason that college tuition costs have far outpaced inflation and gone up twice as fast as medical costs since 1985."
Maybe so. It could also be argued that Government Food Stamps increase the demand for food and thereby increase the price of food because more people with money in their hands (greater demand) drives the price up. If so, it would then stand to reason that ending all Government food assistance would help drive the prices of food back down.
Although this might be true to some small extent, the result of ending Food Stamp programs would be that families with no money at all would starve to death, even if food prices reduced to half of what they currently are.
Families with no money at all cannot buy food no matter how much the price is reduced.
Likewise, if the Government ended all tuition support for students to attend public colleges, this might have the effect of reducing the number of students attending those public colleges, If the Government provided free education for all students who wanted it, this would force private universities to offer better courses at competitive prices, because otherwise they would lose many of their students to public universities. But, you won't find a single conservative arguing for this proposal, because they simply don't like government support for education regardless of how it effects the costs of schools.
Google and read this following argument against student loans, in order to read more right-wing supply-side arguments against Government assistance: "Obama's Student Loan Forgiveness Plan Must Focus on Causes, Not Symptoms."
When the Government provides state education that is subsidized by tax dollars, this reduces the price of government education for everyone. Some students who might have gone to a private university will choose a public one because it's government subsidized and less expensive. According to conservative ideology, that would actually decrease demand at the private colleges and force the privates to lower their tuition.