Grammar is a generating device means "it defines the rules/productions for generating valid strings which collectively form a language."
Analogy-In English grammar statements follow rules as- Subject Verb Object. eg- She eats the apple.
Automata is cognitive device means "it reads the language generated by Grammar using its transitions"
Analogy- You are automata as- eg- "She eats the apple." You will read this and accept it. However, If I say "She eat the apple" , this is wrong as per English grammar and you will not accept this.