They put all the ballots in piles by everyone's first choice. The smallest pile (e.g. least voted for candidate) then loses, and their ballots get sorted onto their second choices. Repeat until you have a candidate with a clear majority, and you have a winner.
This allows people to say, "I want Rubio, but if not, then Cruz, and if not then Jeb" etc. I suspect if this system were in play in both parties now, we'd have different nominees, and they'd be the candidates that the majority can live with instead of being the candidates that most excite the extremes. And the people on the extremes can see that their candidate lost even with people being able to vote for them without 'wasting' their vote.
Yes, in our senate this gets a bit crazy with tablecloth sized ballots (https://upload.wikimedia.org/wikipedia/commons/a/aa/Victoria...), and the first time I voted here I allocated every single preference down to about 160 something with pride. You don't have to obviously, but I chose to.
Our Electoral commission is the most trusted government body in the country, and has maintained a culture of independence and accuracy. We don't have any form of electronic voting, but generally have election results on election night.
Our system has multiple viable political parties! Factions on both the left and right of the spectrum at multiple government levels have successfully won and lost seats over the years depending on their ability to poll within the electorate. This has not resulted in them being regarded as "spoilers" to the main political parties, and has acted at times as an effective check on government policy since it encouraged cross-party negotiation through multiple avenues.
Is our system perfect? No - no system is. But come election night, our representative government actually represents the people. If you got 51% of the vote, then 51% of the population, through some means, selected you as a preferred candidate.