What does Plato mean by claiming that the spirited part of the soul is distinct from the calculating and the appetitive parts?
The first issue one needs to address in considering the Platonic account of the tripartite soul in his Republic is that Plato was writing in Greek, not English. Many of the standard terms used to translate Platonic Greek have a long history in English.
Through the early 20th century, Anglophone (and most European) classicists learned Latin before Greek and then learned Greek through Latin, i.e. when one had difficulty understanding a Greek passage, one would translate it into Latin and then from Latin to English. Most of the early translations of Plato into English were actually English versions of French translations of Latin translations of the Greek. This means that words are used in Platonic studies in specialized technical senses unlike their ordinary English usages.
In Plato, the three parts of the soul are nous (“intellect” or the rational part), thumos (“passion” or will or spirit), and epithumia (“appetite”). The appetitive part desires things like food or sleep. The intellect or rational part reasons. The spirited part or will is the force that enables the person to act, and has a sense somewhat close to our understanding of will power, or self discipline, or motivation.
The tripartite structure of the soul accounts for the experience of our being divided within ourselves -- imagine that your body wants a cookie and your mind says it is fattening. Sometimes your will can aid your mind in resisting your desire for the cookie and sometimes not -- so to fully account for this example we need 3 parts to the soul.