Those of us who grew up in the era where we learned typing on actual typewriters were generally told to use two spaces. I know I was. In fact, I put two spaces in after the periods in this post, even knowing that under HTML the extra spaces will be discarded.
More recently, I worked for several years in an environment where we prepared what amounted to white papers in Microsoft Word for publication. One of the official standards was that we must put two spaces after periods at the end of sentences. This was because (and I agreed with this) it aided in readability of our work. We also set the font size at 13 rather than 12, and added a hair of extra line spacing (like 1.1 instead of 1) again for the purpose of readability. I think these ideas helped a lot -- I liked the look of our published product.
But in a large number of cases, the two spaces after the period are meaningless. They mean nothing when text for web display is entered, because web browsers ignore all "white space" beyond the first space character encountered. In other words, even if you put five or ten spaces after a period, only a single space will be displayed on the web browser. The extra spaces also are not needed for manuscripts being submitted for commercial publication, where the publisher will be formatting the text to "pretty it up" with a selected typeface and justified margins.
Nonetheless, the spaces are still hardwired into my typing fingers. They still show up in emails and word processing documents, and I personally like them -- as a readability aid.