What is going on when the water that is draining from the washer pushes up from the bottom of the toilet in next room?

It means you have a blockage downstream from the toilet. If you can find a full size cleanout in the home, or near the outside of the home, pull the cap and look for standing water. If no cleanouts are readily available, pull the toilet and snake the line with a large augering machine. I would actually just suggest augering from the toilet drain since you will have to pull the toilet to change the wax seal anyway, now that it is leaking. Even if you do find a good cleanout, and snake from there, you will still need to pull the toilet and change the wax ring, as it has been compromised and will allow sewer gas into your home until you change it.

Oh, and are you on city sewer or a private septic tank? If you have a septic tank, I would start by checking to make sure the tank is not full of solids or merely having the tank pumped if it hasn't been done for a few years. Either way, that wax ring will still need to be replaced. Good luck, and please let me know how things work out.