For example in the sentence: "This tells Google how to index the page" what does Index the page mean in the grand scheme of things. Why would a page have an 'index.' What is it useful for?
Google servers are constantly visiting pages on the Internet (crawling) and reading their contents. Based on the contents Google builds an internal index, which is basically a data structure mapping from keywords to pages containing them (very simplified). Also when the crawler discovers hyperlinks, it will follow them and repeat the process on linked pages. This process happens all the time on thousands of servers.
In general, the term indexing means analyzing large amounts of data and building some sort of index to access the data in more efficient way based on some search criteria. Compare it with database indexes.