Why does quantity supplied increase with price in economics?
I believe that I have found the problem in this line: 


  I don't understand why price has to increase if quantity increases.


The concept is actually the other way around.  

Think of it in the same way as the Law of Demand.
If you go to the grocery store and you see a food that you like selling for \$.25/lb, you would buy a whole lot of it before the price rises.
Conversely, if you go to the grocery store and you see a food that you like selling for \$100/lb, you would probably wait to buy this item until it is cheaper or at least buy a small amount of it.
In economics, the price drives the quantity demanded by the consumer.  

Now let us look at the Law of Supply.
Imagine that you are the owner of a company.  You go to the store, and you see that the item you are producing and the similar items produced by your competitors is selling for \$.25.  You would not necessarily want to produce a lot of the product because the margin between the selling price and the production costs (profit) is small.
Conversely, imaging going to the store and seeing that the item you are producing and the similar items produced by your competitors is selling for \$100.  You would want to produce a lot of the product because the margin between the selling price and the production costs is (presumably) large.
In this case, as in the other case, the price drives the quantity produced by the supplier.