Transistor is not switching-Raindrop sensor with LED (Later Buzzer will be used)
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Please use the digital output pin for the module.  

As per the schematics, it is the output of the comparator. You don't need a microcontroller to use the digital output pin.  

Below is an example circuit I recommend. 


The digital output of the module can be connected to the base of the Q1.  
The three LEDs can be replaced by your single LED. Forget the tricolor LED. Please use a simple one for now.   
When there is no water (rain), the digital output of the comparator will be high. This will turn on the Q1. This will turn off the MOSFET M1. Hence the buzzer (LED) will be off.
When there is rain, the Digital output will go low. This will turn off the BJT Q1. MOSFET gate source vorlage is high enough to turn it on. Bizzer (LED) should be on now. 
With a multimeter please check the behaviour of Digital Output pin D0 to make sure it is toggling. Use the available options to tune your circuit.