Why is international trade conducted in the US dollar?

Because without a global currently, international trade would be like barter. Each country can print only its own currently, but has to pay other countries in their currencies. Therefore, it has to buy their currencies.

Imagine stage zero in this game. Each country has only its own currency and the international currency market isn't there yet. France wants to buy a radio from Japan and has to pay in yen. Where can it get yens? Well, only from Japan. What it will pay with? French goods or French currency. But these two are equivalent because Japan needs francs—put aside the euro for the reason below—only to buy French goods, since each country accepts only its own currency.

If the Japanese don't want French wine, the trade will not take place until both sides will find the third side, say the US, that wants French wine and produces movies the Japanese like. Now, they don't even need currency to do this exchange: wine goes to the US, radio goes to France, and movies end in Japan.

The common currency (furs, silver, gold, dollar, euro) helps to resolve this rather inconvenient situation by introducing a currency everyone wants because everyone else is ready to accept it in exchange for goods.

There may be several global currencies, but the more global currencies we have, the less convenient the world becomes, because each currency's relative circulation decreases. Euro is the one; and you will find French firms that pay for Japanese goods in yens. In general, it's possible to find a way to pay in any currency, but it's much less convenient.

PS: The parallel with tourism doesn't work primarily because tourists actually pay in euros and dollars. You would note that in less developed countries locals will accept or exchange dollars or euros, but not rubles or yuans.