When you do something reticently, you act hesitant or shy about it. A contestant in a singing contest might surprise you with her beautiful voice after she shuffles reticently onto the stage.
To be reticent is to hide your feelings or to be bashful, reserved, or quiet. So when you act reticently, you show this reserve in your actions. Although your silent grandfather greets you reticently, he loves you just as much as your enthusiastic, effusive grandmother does. Reticently has a Latin root, reticere, "be silent."