Why does quantity supplied increase with price in economics? The Law of Supply is my worst enemy in economics because I could never truly understand it, and as a result, the stuff I learned after that was built on a weak foundation. The Law of Demand is totally different though, it makes perfect sense to me. I have spent hours thinking about this and I've figured out exactly what I don't understand. 

The Law of Supply would make perfect sense to me if price was substituted with revenue. With each good supplied, revenue would increase in a linear manner. However, as I realized, it is not revenue we are dealing with. The way the Law of Supply works, revenue would increase in an exponential manner if we took a supply schedule and multiplied price and quantity for each price level. 

I don't understand why price has to increase if quantity increases. Shouldn’t the increase in quantity supplied already generate more revenue to cover the extra costs of production? Why should the price be spiked up to further increase revenue? Why wasn’t the price that high in the first place if the good could be sold like that? 

The Law of Supply seems counterintuitive to me in some ways. If you have only a small amount of something to supply, shouldn't you make the price HIGH so that those rare fools willing to pay for overpriced things (the ones at the top of the demand curve) would clear your stock and maximize your revenue? 

One explanation that almost made sense is that the more you tried to produce, the higher the costs of production would get. A producer would have no choice but to raise prices if costs of production were that high. But this only makes sense if costs of production increase EXPONENTIALLY, which I don't understand why would happen with EVERY good! Why is it always assumed that marginal cost is increasing and not remaining constant? 

On a side-note: how would the Law of Supply work in the digital realm, where stock is basically infinite? What would a supply and demand graph look like there? 

On a side-side-note: why are the supply and demand graphs really considered CURVES when they are almost always represented by lines? 

Maybe there's something fundamentally wrong about my understanding of this. It's supposed to be easy to understand, but for me it's not. I would be so grateful if someone could clear this up for me!
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.