Reddit

An Application Programming Interface (API) is basically a way to let apps communicate with the server (Reddit, in this case) directly, in ways that are faster and more efficient (but less user friendly) than the site page.

For apps, having an API for Reddit meant that they could just ask Reddit directly for a list of posts, and submit any interactions without having to pretend to be a user and "read" the site, which might break as soon as Reddit changed something, like redesigning the website.

For Reddit, that meant that it didn't need to send the apps a whole page, and waste valuable bandwidth/processing power. It could just send the relevant bits the apps needed, and save resources/cost.

Running an API isn't free, since it is basically another part of the site to maintain and deal with. Reddit officially claims that the costs associated with the API (from sending the data to apps) are too great, which why they are clamping down on it, and increasing costs so significantly (although it is worth noting that the numbers are orders of magnitude higher than the API costs for similar services, like Imgur, and higher than what Reddit's own costs would be, so take with a mountain of salt).