Why is Wake-On-LAN not working on my MacBook Pro with LogMeIn?

It is not possible for the Macbook to go out of sleep when the lid is closed and no external display and input devices are connected. Your best option therefore is to keep the lid open.

The checkbox for wake-on-LAN only allows the ethernet interface to respond to what's called a "magic packet" that will then cause the system to wake up. I don't know what the LogMeIn dashboard looks like, but if it's just a status report for the other computers on your local network, then it is reporting the state of your MacBook accurately. If the Dashboard has the capability of sending the magic packet, then that is a step you will need to execute before the machine comes back online.

If it doesn't do this, you'll need an extra tool to send the wakeup call.

Just making sure, you're doing all of this from within your local network, right?