Another way to look at the phrase "to care and not to care" is as an invitation to challenge habitual relationships to experience. Our conditioning tells us to care about (make important) things that are pleasant, make us look good, and otherwise confirm or prop up our sense of identity. We also care about things that threaten this sense of identity. On the other hand, habitually we do not care much for things/people that aren't connected to us or are otherwise neutral to us.
Spiritual practice takes a different, somewhat orthogonal approach. We train ourselves to care about (notice, make important) the aspects of experience that are uncomfortable, unsatisfactory, or bring a feeling of contraction. We learn to care for what hurts. And conversely, we learn to care less about fame, fortune, pleasure, and other things that are mere temporary hits. Interestingly, over time, these together create a new caring for what is peaceful, tranquil, or sublime.
The emergence of this new type of care is surprising to some, but also tends to feel familiar. Many practitioners discover that "neutral is more pleasant than pleasant." To find the peace in a given situation, it is necessary both to care and not to care -- to be connected and engaged, but with an openness to what will unfold.