Part (a)
The equilibrium price occurs at the intersection of the supply and demand curves. See the diagram below to see how this looks. Ignore the red line and green lines for now. Those will apply for part (b).
If the price was below equilibrium, then we would have a shortage and it would lead to a pressure to push prices upward. If the price was below equilibrium, then there would be a surplus or oversupply, leading to a downward pressure in prices. So inevitably the price lands on the equilibrium if given enough time and there are no external forces acting on the market.
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Part (b)
Let's say we have two people labeled person A and person B. If person A gets a higher education, while person B does not, then person A is likely to earn more money. Person B will see this and want to may want to earn more money as well, so that makes their demand for higher education go up. While of course this rule doesn't work for everyone, in general we'll have enough people to effectively push the demand curve to the right while the supply curve remains where it is. Check out the diagram below to see what I mean.
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Part (c)
Refer to the diagram in part (b). We have a new demand curve which intersects the supply curve to produce a new equilibrium point with price P1 and quantity Q1. So effectively the tuition has gone up due to higher demand.
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Part (d)
If a government provides a subsidy, then this would lower the cost of education to provide people with more opportunities to improve their earning potential. The subsidy would provide a portion of the tuition which gives the illusion of the tuition price being lowered. In a way it's like getting a grant from the government to then pay that money toward your tuition.