I have checked through music notation lists and have asked a number of musicians but nobody knows what this symbol is actually supposed to mean.
It's a simile. There are a few different types of similes and this one means "play the last notated measure again". So in this piece you will end up playing the measure before the simile marks 3 times, then play the next notated measure.
It's pretty much a very shorthand way of saying "Play what you just played again".
It is short-hand for "play this measure the same way you played the previous measure". Sometimes it is called the "repeat bar" symbol. It is frequently found, for instance, in fake-book charts and in notation for the "rhythm section" in jazz, meaning percussion, bass, piano and guitar (with guitar, particularly for indicating that you are to repeat a certain strumming pattern for chords). Sometimes you find a numeral over the symbol, meaning "play the same one-bar pattern for 2 measures", or 3 measures, or 4, etc.
Repeating what everyone else said: that symbol means repeat the last bar. Periodically, there are also ones that mean repeat the last two (or however many) bars. I don't remember off the top of my head what made that one look different, it was very similar, but I believe there was a number printed that helped with that conclusion.
They can't just put in normal repeat signs because the melody does not repeat, and that would mess up measure numbers, but it saves ink and your effort once you're used to it, to just say "repeat" like that.