Go to System -> Remote Desktop Connections and check the box to enable incoming connections.
On your router, forward port 3389 to the machine you want to access.
On any client device running Windows (linux/mac can use FreeRDP, for which there are numerous wrappers) connect to your home's Public IP, and log in with your computer's usual username and password.
For the average home user this is all the setup that's needed, and it doesn't involve handing over your computer's security to a third-party company that is now a giant target for these sorts of attacks. :)
I think the big advantage that LogMeIn and GoToMyPC have in the casual realm though is their simplicity. In particular, they entirely dodge the need for the user to know anything technical at all, including port forwarding on the router, which is the bigger technical hurdle here. You just install their program and it does the rest for you. Port forwarding seems like child's play to you or me, but they have a huge market in folks who didn't even know you could log into your router and change settings. Or that even know what a router is, despite owning one.