If this starting problem happens only on very wet conditions, if not then the problem is the fuel injection system.
The reason why the vehicle starts but the engine backfires.
(1) Moisture in the distributor cap.
(2) Problem with the fuel injection system.
Car troubleshooting
Moisture in the distributor cap
As indicated, backfiring only occurs in wet conditions.This is a telltale sign of moisture on the inside of the distributor cap. The moisture is condensation which results from the humidity in the air. This seriously interferes with the ignition system's ability to operate properly, causing this vehicle to backfire.What to do?
It isn't much you can do in this situation except, in the future, park this vehicle in a dry place (i.e. a garage) when you anticipate rain. This should help.
Problem with the fuel injection system
It could be that the fuel injection system or one of its many electronic sensors is faulty. This could possibly cause backfiring.What to do?
Fuel injection systems are complex and hard to diagnose. If this vehicle is not under warranty, take it to a mechanic that has expertise in this particular make/model. We highly recommend you get a second opinion if the first mechanic suggests that you need a new fuel injection system. Very often, the problem is not the fuel injection system itself but rather one of its sensors. A skilled mechanic will be able to determine if all the sensors are working properly.