Typhoons are the Pacific equivalent of hurricanes, great big storms that only hit occasionally, almost always in the August-October typhoon season, and Taiwan gets an average of 3-4 per year. If a typhoon hits squarely where you happen to be staying, you can expect to be washed out for a couple of days, possibly even briefly stranded if you're out in the countryside if there are landslides that take out roads and railways.
However, between the typhoons (which is 95% of the time), you'll be absolutely fine, with clear fall skies and the occasional random sprinkle. Here's a rainfall chart:
As you can see, outside the sparsely populated east coast (=Hualien), it rains a lot more in the summer than it does in either of Sep or Oct.