Typewriters used monotype lettering, meaning, no matter what the letter, every single letter took up the same amount of space. So an ittie bittie lowercase "i" needed to be just as wide as a monstrous capital letter "M" (see fig. 2).
What happened then was computers. Computers started getting smart. Not only were they able to use type that wasn't hampered by physical limitations, but they also knew how to give each letter the spacing it needed for optimal reading. They didn't need us to tell them about the extra spaces we had been using after a period. It already knew how much was necessary and said in a condescending kind of way, "Yes. I know."
Problem was, the people who learned how to type on typewriters (meaning: us old people) kept teaching new typists (meaning: those of you who grew up learning to type on a computer keyboard) that double-spaces after periods were the way it was done, mainly because we didn't know any better.
|fig. 3 Typographers should avert their eyes from ¶ 1|
If you want to fight this change to the death, here's a few good lines taken from the link above:
Using a single space means that you understand that technology has changed since the decades ago when you first used to type. A single space means you realize not everything your teachers taught you in high school still holds true. A single space means you have respect for the journalists and designers who are working hard to take those extra spaces out of the drafts you're sending us.
And who wouldn't want to show their high school teachers that they know better?