The concept of virtual representation was meant as a rebuttal of the American colonists' claim that they were being taxed by Parliament without being represented in Parliament. The idea of virtual representation was that the members of Parliament were representing the colonists even though the colonists could not vote for them. Parliament, it was argued, had the colonists' best interests in mind and was, therefore, representing them.
The colonists did not like this idea. They wanted actual representation. They argued that no one could truly represent them unless they had the right to vote for or against that person.
Today, we still have virtual representation in some ways. We tax minors who work, for example, before they are able to vote. We say that their interests are being protected even though they cannot yet vote. The problem in colonial times was that the British wanted to treat all colonists in this way.