Supply and Demand: Why should lower costs lead to a rise in supply? I know that the conventional explanation linking fall in cost of production and rise in supply is that a fall in cost of production of A means that it is more profitable to produce A. Thus, suppliers will be incentivised to produce more of A, hence rise in supply of A. 

My question is - why should suppliers even want to produce more? Can't they just enjoy the higher profits attained automatically through the fall in COP? What exactly is the explanation for them wanting to produce more? 

This is the missing link in many explanations I see and I hope to clarify this, thanks!
You can think about this using the basic theory of the firm that we teach in principles courses, such as this one.  The specifics of the graph will change depending on the assumptions of the market it represents (perfect competition, monopoly, etc), but the story should be the same.

In brief, each individual firm makes a production decision so that marginal cost (MC) equals marginal revenue (MR), where "marginal" means the next unit of production.  Keep in mind that a firm is always trying to maximize its profits.  Therefore, if MC > MR (the cost of the next unit of production exceeds the revenue of the next unit of production), it means the firm could increase those profits by producing one unit less.  If MC &lt; MR, then they could increase profits by producing one unit more.  Only when MC = MR can the firm no longer increase its profits by changing the quantity it produces.

If anything happens in the market to lower production costs, it means the firm's old production decision leaves them with MC &lt; MR, and thus they should produce a higher quantity.  Continuing to produce the same quantity would mean they're earning lower total profit than they could be.

Of course there are a lot of simplifying assumptions in this model, which is why it's one taught in principles classes.  But that's the foundation of the idea you're asking about, I think.