Some white blood cells, specifically natural-killer cells, can target virally infected cells that have a major histocompatability complex displaying an epitope that the body deems to be non-self. An epitope is a small portion of protein that can be thought of as a cell’s ID card that shows what sort of stuff is being made inside that cell. Viruses inside the infected cell make proteins, which are cut up and displayed by the MHC-1 (which are found on all nucleated cells in the body). The NK cells recognized the viral epitope as foreign and secrete the cytokines necessary for that cell to undergo lysing.
Theoretically, if only a few cells were infected by the virus the immune system could clean it up by killing all those infected cells (This happens on a regular basis. Some cells are infected and killed by the immune system. Such a case would likely be considered subclinical, with no symptoms/evidence of being sick). However, someone that has fallen ill and has symptoms due to a virus likely has more that just a few cells infected, and would make it nearly impossible to eradicate all the virally infected cells.