I looked around on the internet and found this simple explanation of how it works on The Panhead and Flathead site:
"The way it works is this: To downshift you pull in on a lever on the handlebars (usually a modified British compression release). That raises the shifter rod on the transmission and engages the downshift portion of the mechanism. Push down on the clutch pedal which disengages the clutch and shifts the trans into first. Once it's in gear you release the lever as all it does is move the shift rod into engagement with the downshift cogs on the Speedy Shift. After you release the clutch and start moving every time you push on the pedal it up shifts at the same time. To downshift just pull on the handlebar lever and every time you push on the clutch it shifts down.
I know it seems complicated (and it is) but back in the early 50's they were the hot ticket to go fast. I have ridden one before, and once you get used to it you can shift incredibly fast. As a side note, you had to have your transmission set up perfectly as it was very easy to gut one using one of these."
David Mann drawing of a Speed-E Shift on a Shovelhead..Pretty accurate