I'll try to explain this as simply as I can, although I might be skipping over some mumbo-jumbo unimportant details--forgive me if I do.
The Bayer pattern is reliable, but not without issues. One problem is that fine repeated patterns (like a window screen or fabric or a shingled roof) can cause moire pattern distortion. This is basically weird wavy lines where there shouldn't be any. Another problem is that straight diagonal lines tend to have a "step" or "Lego" effect instead of a smooth transition. These problems are fixed with an anti-aliasing filter (also called optical low pass filter), which blurs the image slightly.
Interestingly, the more resolution a camera has, the less of a problem these things are. Some cameras with a Bayer sensor (usually those with at least 24 megapixels) don't have an anti-aliasing filter.
Fujifilm took a different approach. They wanted to do away with the resolution-stealing anti-aliasing filter, so they rethought the color pattern on the sensor. They used a more random design with 55.5% green sensitive sensor elements, 22.25% blue sensitive sensor elements and 22.25% red sensor elements.
The X-Trans color pattern made the anti-aliasing filter unnecessary because it is not subject to moire pattern distortion and because diagonal straight lines don't look like steps when viewed close up. It's all thanks to the semi-random pattern found on the sensor. This allows for potentially sharper, more detailed images.
A positive side effect to the X-Trans sensor is that, because it has more green "pixels" than a Bayer sensor, it is better at low-light. X-Trans sensors have a one-stop high-ISO advantage over similar Bayer sensors, and even match some full-frame sensors.
I used DPReveiw's Image Comparison Tool to compare three different Bayer sensors (with similar resolution) to Fuji's X-Trans sensor. The top two cameras--the Fuji X-Pro1 and the Pentax K-50--both have 16-megapixel APS-C sensors. Notice how ISO 6400 on the Fuji looks similar to ISO 3200 on the Pentax. The bottom two cameras--the Canon 1D X and the Nikon D4S--are both full-frame cameras and they look similar to the Fuji at ISO 6400. The noise produced by the Fuji sensor is less saturated, which makes it appear more film-like in my opinion.
This is not to say that the X-Trans sensor is perfect. It's a lot more complicated to process the files. Adobe had had trouble converting X-Trans RAW images for several years. The sensor seems to struggle with super vibrant colors, and tends to shift vibrant red to more of an orange. These are the trade-offs.
None of the things mentioned are big deals. X-Trans sensors aren't better than Bayer sensors. They're just different, with different positives and negatives, none of which are major. I wouldn't base a camera buying decision on this alone.
What made me choose an X-Trans camera was 1) Fuji's excellent lens lineup, 2) the camera's high-ISO performance, and 2) Fuji's brilliant in-camera JPEG processing, which, when your RAW workflow is as backed up badly as mine, you really need. Also, cost and camera size were big factors.