For diodes to conduct, the voltage at positive terminal should be greater than voltage at negative terminal by threshold voltage of diode.
So diode conducts only when you make terminal A/B to be low (logic 0) as DA/DB are at higher potential. When this happens,diode acts as short circuit and output is 0
When a high voltage is given at A/B, respective diode are reverse biased and diode acts as open circuit, so output will be high.
This is a AND logic gate