I am a Christian, but this question has been troubling me for a long time. If all the world's animals dispersed from Mt. Ararat, it would stand to reason that there should be no animals in the Americas today. This cannot be explained by plate tectonics, a process far too slow to be of any aid. Or perhaps the flood did not affect the Americas? Or could they have crossed over the Aleutian islands, on the land bridge across the Bering Strait through which humans first arrived in the Americas? If so, when did that bridge disappear and how did it disappear so quickly?
Usual preface that seems necessary to head off debate on such questions and answers - this isn't endorsing or denying the YEC view, just answering how various YEC groups would answer the question. Whether the YEC'ers are right or wrong is completely off-topic, as has the long-standing policy been.
While we can't answer this definitively, several possibilities have been proposed by those that hold the literal Young Earth Creationist view, which includes a literal global flood,
ChristianAnswers.Net, for example, while firmly stating that we don't know for sure, offers the possibility that land bridges once joined the various continents. This isn't too far off from the old-earth view that people crossed to the Americas from Asia via a land bridge that formed during the ice ages.
Answers in Genesis, also prefacing their article with a "this is what we think, but we don't know for sure" disclaimer, discusses the fact that we know very little of what the earth was like immediately following the flood. Continents may still have been shifting, there may have still been major upheavals and changes not recorded, and the re-colonization of the world may have been possible because things were still connected - again, land bridges.
Some others, including this blogger think that Pangaea was real, and was broken up in the days of Peleg. It's not a widely accepted interpretation of the text, but this blogger isn't the first that I've seen use it:
There is an obscure passage in Genesis that is generally overlooked, but holds what I believe to be great insight into our early world after the flood. Genesis 10:25 states “Two sons were born to Eber: One was named Peleg, because in his time the earth was divided”. I believe this passage describes a time when God supernaturally separated the continents one from another. The Bible says that Peleg (which means divide) was born 100 years after the flood and lived for 239 years. The time frame for the continental split would therefore have been 100 years after the flood until 339 years after the flood. The division is said to have taken place during his lifetime, but doesn’t say how long it took.
Cretation.com goes as far as to say that the bigger problem is for the evolutionist, citing the fact that so many similar species exist on each continent. They argue that if the evolution theory were true, you'd expect far greater differences between life on different continents. They argue that the fact that so many similar and even same species exist on the various continents is evidence that they did not evolve in isolation, but sprang from the same Biblical kinds in recent history.
Another creation.com article offers an alternative suggestion, or perhaps supplemental explanation to land bridges: Natural rafts.
Another explanation which is gaining increasing support is the rafting hypothesis.
Interestingly, the potential for dispersal of plants and animals across large stretches of water by natural rafts has been accepted by evolutionists for many years. Professor Paul Moody of the University of Vermont argued, “In times of flood, large masses of earth and entwining vegetation, including trees, may be torn loose from the banks of rivers and swept out to sea. Sometimes such masses are encountered floating in the ocean out of sight of land, still lush and green, with palms, twenty to thirty feet [7 to 10 m] tall. It is entirely probable that land animals may be transported long distances in this manner. Mayr records that many tropical ocean currents have a speed of at least two knots; this would amount to fifty miles [80 km] a day, 1000 miles [1600 km] in three weeks.”1
The Old Earth Creationist view of Hugh Ross believes the flood was limited to the whole ERETS (country) and limited to animals who were NEPHESH (emotional). The NEPHESH means that those animals with emotions could be corrupted by hanging around with evil men. Think about it, a bad guy can make a pit bull evil, but he probably cannot corrupt a pet fish or lizard very much-- thus, in the OEC view, only the NEPHESH creatures needed to be destroyed, and only those who'd associated with man. Thus, the million or so beetle species in South America (or areas not yet populated by modern humans) didn't need to be destroyed.
David did an excellent job explaining the "how" as viewed by major biblical literalists. I also being a biblical literalist would like to point out that in Genesis 2:19 God brought animals before Adam, and in Genesis 7:9 we see that the animals came to Noah. Obviously it was something God did, so it was a miracle. I'm not aware of any denomination that claims God stopped working miracles at the flood, so I think it's a reasonable assumption that since the flood and all the events leading up to it were a miracle, how the animals disbursed across the globe was one as well. That is after all what he commanded men to do(and when we didn't He stepped in and confused the languages).
We also see in the New Testament where men were "translated" from one location to another. So bottom line is, could there be a natural means by which every animal got where it was supposed to be? Sure, but it doesn't say. God intervened to get them where He wanted them to be, it was a miracle, the only question is to what extent and how much of natural means did He use. The scriptures are silent on this.