If you have ever flown on a true budget airline, you will see some parallels within the Ouigo details. For example, the main train station is not in Paris. In fact, it's about a 40 minute commuter rail trip outside the city, by Disneyland Paris, at Marne-la-Vallée.
Also, don't think the 10 Euros will get you anything more than a seat. If you need to charge your devices, that's an extra 2 Euros; some extra baggage will add 5-10 Euros to the ticket price. Make sure you bring your own nibbles for the roughly 3-hour trip south since there is no cafe car when your feeling peckish. Obviously it will be more economical to pack passengers in like sardines.
As expected, the cheap-y prices are a supply and demand equation. The faster the cheap seats sell out, the higher the prices go. A 10 Euro ticket can easily turn into 30 Euros if you wait too long. So if you can plan ahead, you might be able to snag a seat for a few bucks. We would also recommend brushing up on your French, since the website seems to be "'sans Anglais."