Conservatives say that we need people to be “responsible”, to take “responsibility”. This is not an equivalent, especially in that context, of “autonomy”. The stress on “responsibility”, for conservatives, is there to justify, strenghten, deepen and expand traditional hierarchies.
For liberal humanists, I believe, individuals are only “responsible” before the law of their actions. In the civil world, they should be “autonomous”.
The care for other people comes from the ethics that, again, human welfare is the highest good. At the level of society, it translates in utilitarianism: “the greatest amount of good for the greatest number of people”.
You care for yourself, for your family, and your fellows. This is in your own personal interest too. Parents who don’t take care of their children, will likely not be taken care of by their children when they will be old (especially in open western societies where people have the choice to do so). People who are nasty with others will likely be treated badly in return one day or the other. Same for the environment, btw.