If you do something openly, you don't try to hide it. Some countries allow their citizens to openly criticize the government, while other countries harshly punish their critics.
There might be some things you prefer to keep to yourself, or to whisper about with your best friend. Or, maybe you're a person who prefers to live your life openly, being direct and straightforward about everything. Talking openly about your feelings can be hard, but it usually makes you feel better. Openly comes from the Old English openlice, "clearly or unreservedly."