dual citizenship

Can someone explain to me how you properly travel with dual passports? I am a United States citizen and I want to apply for my German passport. I have to show my US passport upon leaving the United States and show my German passport upon arriving in Germany. Do they look for the stamp in the United States passport? Then when I leave Germany I show them my United States passport right? Then when I go through customs in the United States do they look for the stamp when I entered into Germany? What are officials looking for when using both passports? Also, what benefits do I have by holding my German passport? Thank you so much.

Your German passport won't be stamped. neither will your US one.

They don't "look" for anything.

A german passport, like any EU passport allows you to live and work anywhere in EU without any visa.

No you are not suppose to carry both the passports with you. When your German passport is required give that one, otherwise when US passport is required present that one. Its just simple