The functioning of amino acids as a root stimulator is a highly complex and comprehensive story and the exact answer to these questions can be found in a variety of biochemistry literature.
We will try to describe it briefly. In living beings (including plants), the 20 standard N-containing amino acids are used to build proteins (and other bio-molecules) and they can be oxidised into energy-rich compounds. In fact, life is not possible without proteins and, therefore, amino acids.
They play a part in all life processes related to maintenance, growth, vitality and reproduction. The same applies to plants (plant roots). A plant that receives a proper supply of amino acids in addition to basic nutrition and water will be vital, grow strongly, have a healthy green colour, produce more flowers, have more resistance to diseases and stress and, therefore, higher and better production.
Because all 20 N-containing amino acids are involved in so many essential processes, which are also interdependent in many cases, it is important that the plant has all 20 N-containing amino acids at its disposal. A shortage of one or several amino acids may already limit a chain of processes, just as a chain can break at the weakest link.