Independent events are events that do not affect each other. This means that the outcome of one event does not depend on the outcome of the other. In the case of rolling a die we notice that the fact that we get a 4 in the first roll does not affect the number that we are going to get in the next roll. Therefore, these are independent events.
Mutually exclusive events are events that can't happen at the same time or events that if one of them happens then the other can't occur. Rolling a four in a die and then rolling another four are not mutually exclusive because the fact that a number occurs in one die does no affect the occurrence of the other.
