The drain in the laundry room is a great thing to have, it is there incase of flooding caused by your washer or the hose supplying water to your washer should break. The water goes into the drain instead of flooding your entire house.
It should connect to your main sewer line.
On a side note if you just bought this house it is time to get ahold of your realtor and speak to him about questioning the sellers, if they sold you this home knowing that it had a defect such as this and didn't disclose it they could be responsible for fixing it. IF they had a sewer flood the room and took care of it improperly they can't just pass the clean up along to you.
You need to do some sniffing research. Is the smell coming out of the drain also? Can you tape over the drain and see in a few days if the smell is lessened? Do you have carpet in the laundry room? If it has gotten wet it might be the culprit, can you lift up a corner and look for mold? Can you totally isolate the smell, I mean get down on the ground and sniff like a dog, mark with tape exactly where you smell it, perhaps using a lemon in between to clear your nose so you don't get used to the smell.