You cast the card from exile. Each land card exiled this way will remain exiled.

You must follow the normal timing restrictions of the exiled card.

If the card has {X} in its mana cost, you must choose 0 as its value.

If you can't cast the nonland card (perhaps because there are no legal targets available) or if you choose not to cast it, it will remain exiled.

If there are no nonland cards left in that library, the opponent will exile all cards from their library, and Stolen Goods will finish resolving.

If you cast a card "without paying its mana cost," you can't pay any alternative costs. You can pay additional costs, such as kicker costs. If the card has mandatory additional costs, you must pay those.