The quote comes from The Four Loves, a book published in 1960 in which Lewis explores different forms of love, including affection, friendship, romantic love and charity.
Lewis argues that love involves accepting the possibility of pain. Caring about someone means allowing their actions, choices and absence to affect the heart. That is not a sign of weakness. It is part of what makes emotional connections meaningful.
The deeper lesson is that avoiding every possible hurt can also mean avoiding the experiences that make life richer.