Fish don't go out in the rain because they're afraid of getting wet! :)
It isn't so much an issue of whether or not it's raining, but rather how this changes barometric pressure. Rain does affect how a fish behaves, but this doesn't directly correlate with rainfall.
A storm front is an area of low pressure. Fish can sense this change using their lateral lines. These nerves are used to detect movement in the surrounding water. A drop in pressure means two things to a fish:
The water will be stirred up by incoming winds, making it easier to feed, especially near the surface of the water.
Once the storm hits, the water will be moving too much for the fish to feed.
The fish will be more likely to bite just before a storm because they know it may be a while before they can eat again. They'll stay out while it's raining as long as the conditions remain mild, but they'll be moving around a lot to get as much food as they can. Professional fishermen take advantage of this:
This agitation may also give you an advantage in bodies of water that are normally clear: By stirring up mud, the fish won't be able to see you. This means it's much more likely to bite a lure, and you can get away with casting closer to your boat than you would when the water is still.
Posted 2119 day ago