Even you turn off the computer it still has circuit standby using very small current, just like the TV, even you turn it off you will still see the LED light on. It allows you to turn the TV on again with a remote. You can try to see the back panel of a computer when it is off, you are likely able to see some light on.
In able to turn on a computer from home using Logmein you do need to leave your modem and router on. Make sure the LAN on your office computer under BIOS is enabled. When you log in to Logmein account at home you will see an green icon for you to turn it on. That's it. It works well for me so far.
