In a two-player game, assuming each player eventually exiles a nonland card, your opponent will choose one of them, and you'll be able to cast the other one.

A spell you cast this way can be the target of a later spell you cast this way. However, permanent spells cast this way won't resolve until you're done casting spells, so the permanents they become can't be the target of spells cast this way. For example, if you exile Twincast and Lightning Strike, you can cast Lightning Strike and then cast Twincast targeting it, but if you exile a creature card and an Aura card, you can't cast that Aura targeting that creature.

If you cast a spell "without paying its mana cost," you can't pay any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, those must be paid to cast it.

Cards not cast this way, including any land cards, remain in exile.

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

All the spells you cast due to the triggered ability are cast during the resolution of that ability. You can cast them in any order.

In a multiplayer game, you choose an opponent to choose a nonland card exiled with Plargg and Nassari.