I got a couple of these from Amazon a while back and used them in multiple small projects.
However, no matter what projects I did, the LEDs never turn on, and the switch has absolutely no effect.
On all other shields and proto-boards I've tried, whenever there is a switch/button on the board, pressing it always restarts the board.
Yes, I did try looking for info on the HiLetgo website, but there is little/no information there.
Any leads/hints as to what the button and LEDs are for?
The LEDs and switch are not connected. You can connect them as you like.
original Adafruit board:
looks like yours is moved around a bit but you have to actually connect the buttons and LED to make them work. It was designed for prototyping (to give you more options) not plug and play.
So to make it work, you can jump the output to the connection but the power and ground must be connected and use a resistor for the LED if it doesn't have one. It may require soldering some jumper wires. I suggest 22 ga. Solid wire is fine.
Thank you gentlemen ... 'naive' of me to assume they were connected. 
