I'll try to explain it in brief:
The brain has different regions used for different purposes. The brain is NOT symmetrical in how it divides up these different purposes. For example, in most human brains, the regions that control speech (Broca's area, which controls the ability to produce speech, and Wernicke's areas, which controls the ability to understand speech) tend to be found on the left half of the brain (the left hemisphere).
The way the brain-body connection is "wired", the signals controlling the body are crossed: the left hemisphere controls the right side of the body while the right hemisphere controls the left side of the body. Therefore, in the majority of humans, right-handed people are powered by their left hemisphere. As you'd expect, left-handed people in general are powered by their right hemisphere (although this gets pretty complicated and there are many exceptions).
Now, because of the above, and because the 2 hemispheres each have their own specializations, it is possible and perhaps probable that if a right-handed person is therefore powered by their left hemisphere, and language is located in the left hemisphere, then right-handed people have a better felicity for language-based things, simply because their left hemisphere is more dominant (as manifest by handedness). Other specialities found in the left hemisphere include areas for speech structure, mania, and rationalization.
Similarly, left-handed people are mostly powered by their right hemisphere. It turns out, the right hemisphere seems to contain the brain regions dedicated to spatial comprehension, timing, sequencing, geometry, picture-storage, and music-based sensory. <-- One can guess that because many of these right hemisphere traits are associated with art/music, then perhaps many artists/musicians are left-handed.
How do scientists figure out which hemisphere controls what? Well, there is a part of the brain called the Corpus Callosum, which connects the left and right hemispheres so that they can communicate with each other (this is why the image we see through our eyes is not 2 separate "windows" but rather one unified picture - - - because the hemispheres are able to communicate with each other to create a single "window").
Now, in some patients with severe seizure disorders, the only treatment for them is to cut the Corpus Callosum so that they can still function but their seizures won't be able to cross over to the opposite hemisphere and screw up those functions. In these patients, whose Corpus Callosum has now been cut, their left and right hemispheres cannot communicate with each other. These patients can still get on with life (since most daily tasks don't really require both hemispheres teaming up together anyway), but scientists are able to do all sorts of visual, musical, and lingustical tests while they visualize the brain on PET/MRI scans to see which areas light up during certain tasks. Pretty cool stuff!