I am learning about shifts in supply and demand. Things that shift supply include things like improvement in production technologies. But why would a firm want to increase their supply if they are just going to sell more product for less at the new equilibrium ?
The new "equilibrium" would have been at a lower price on the assumption that the firm has full Monopoly over the market (and hence cannot instead just compete with some other firm for a larger share of market demand) and demand remains constant.
Moreover unless you know about the elasticity of demand (how sensitive people's wish to buy is with regards to commodity price) you cannot be certain if selling a significantly larger amount at a slightly smaller price might in totality fetch the producer a larger sum of money.
A profit maximizing firm will expand output until marginal return (MR) equals marginal costs (MC). Intuitively, if the revenue of an additional sale exceeds the costs of the sale, you should go for it. On the other hand, if the additional costs are higher, you should obviously refrain from making the transaction. Therefore, a firm will optimally expand or reduce output until MR = MC.
So when costs are shifting down (but the price for the products stays the same for the short term), the last unit is being sold with a profit. Therefore you should expand output until marginal costs equal marginal return again.
What you should do: Get a standard textbook and look up the behavior of firms in two cases: Perfect competition and monopoly.
You will see that the tradeoff you were referring to is only relevant for the monopolist. With a lower price he may sell additional units but he will also receive a lower price for all the units he sold beforehand. Why is this only relevant for the monopolist? Hint: The difference between the two cases (monopoly and perfect competition) is that a perfect competitor faces a horizontal demand curve while a monopolist faces a demand curve that is downward sloping (because it is market demand!). You will see that marginal return will differ in the two cases.
This is what Harman Deep is referring to: The "downward slopiness" of the demand, i.e. the elasticity of demand, is crucial for the problem at hand.