Non attachment is not, not caring. Non attachment is letting go. In The Little Prince, the story says that if you want to know if someone is really yours, let them go. If they return, they are yours, if they don't, then they never were.
At a certain level non attachment comes fairly easy. It comes with the recognition that nothing is really yours, not even your own body, or the thoughts that randomly come and go. If you don't own your own body, how much can you own another?
We are all on loan to each other. Our bodies are the vehicles loaned to us. We are not the lender. Well, if you are enlightened, perhaps you would say that we are both the lender and the receiver. But from the body/mind's normal perception, we are on loan. We will die!
If you have a child, you will get to practice non attachment. At some point you must release the hand, or suffering will surely come home to roost. When you let go, it doesn't mean you don't care. It means you've grown up.
As long as you hold a leash, they will suffer and you will suffer. To be available is different from holding on. A leash creates enemies, being available creates friends.
Non attachment also comes with trust. If you know there is a greater source than your personal self, or a random universe, if you know that nothing happens by accident, then non attachment is easier.
Consider that wherever you have attachment, you lack trust. A daily meditation could easily be noticing attachment and looking for what it is that you are not trusting.
At the peak of consciousness, one might be able to say, "I trust in that which is." What else is there?
When you really look deeply, you don't know where you are, who you are, or what you are. Science can't explain it, but as it ponders more and more the abstract, things such as intention, synchronicity, and consciousness, it's getting closer. One can certainly love science, but not be attached to it.
Everything must be taken as provisional, except the One -- the unnameable, mystery, the no thing that supports what is. Krishnamurti said it so well in his book, "Freedom From the Known."