Can someone explain why 'He cooks a potato' is also correct?
Because there's no explicit subject and you can't tell if it's "He cooks a potato ->Lui cucina una patata" or "She cooks a potato -> Lei cucina una patata"
How could we know, that it was a she? I wrote he/she.
Why not 'cook a potato'?
So it can never be "I"?
No,it would be "CucinO una patata"