1) convenient defaults. People trade in dollars for the same reason they speak English - because it's mutually convenient between two different countries with their own language and currency. This is partly due to sheer size and partly due to successful cultural export.
2) clearing and settlment. This is not well developed globally, so in practice a lot of smaller countries, foreign banks, and multinationals keep US bank accounts so they can more easily handle payments. This system plays by US rules, which is why the EU is constructing a parallel one to trade with Iran.
3) "eurodollars". I don't fully understand this aspect but apparently this market is coming under regulatory pressure.