What is the intuition/motivation behind compact linear operator. Compact Linear Operator is defined such that the operator will map any bounded set into a relatively compact set. Why is this property so special that it can be named as "compact"? Does it share some similar properties as compact sets? What is the motivation to define and study such a set? 

The motivation to study compact operators is that a lot of the operators that we are interested in are compact.
The reason the property of being compact is so special, as you say, that it deserved to be given a name is that it has many useful and interesting consequences, of course.
In fact, you can generalize this: for almost all X and Y, the motivation to study objects of type X which have property Y is that lot of objects of type X that we are interested in have property Y.
