It means “immediately” or “from the very beginning.” It’s a term from racing. The place where the horses start is called the gate. “Right out of the gate” means “immediately after leaving the gate,” so we use that to mean “from the very beginning “ or “immediately.” It’s not incredibly common, but you do hear it sometimes. For example, “I could tell right out of the gate that she was going to be trouble.” That means “I knew as soon as I met her that she was going to be trouble” or “I knew from the very beginning that she was going to be trouble.”