Responsibilities go hand in hand with contracts. Contracts are covered in the very end of this teaching material.
Objects that act as servers manage a certain amount of responsibility
Objects that manage too many responsibilities tend to be too complex. It is almost always better to distribute the responsibility on several "smaller objects". - The same is actually true for operations.
Preconditions and postconditions is an important topic in the lecture about Contracts and Assertions. See here.
You should care about the responsibilities of both objects and operations
The distribution of responsibilities will become a major theme later in the course