I hear people writing these programs all the time and I know what they do, but how do they actually do it? I'm looking for general concepts.
Technically, screenscraping is any program that grabs the display data of another program and ingests it for it's own use.
Quite often, screenscaping refers to a web client that parses the HTML pages of targeted website to extract formatted data. This is done when a website does not offer an RSS feed or a REST API for accessing the data in a programmatic way.
One example of a library used for this purpose is Hpricot for Ruby, which is one of the better-architected HTML parsers used for screen scraping.