Why is caring an important skill to leadership? 

If you want the people you lead to care about the things that important to you, like deadlines, budgets, organisational goals, etc. You need to first invest your time, each day, each week, to really get to know the people you lead and care about whats important to them.  Remember the little details, their name, their spouses' name, their pet's name, their hobbies, etc.   

When you start to care, then big things will start to happen.  I recommend that you go read 'A Busy Leader's Guide to Caring Leadership' by Joe Deiciano and watch our podcast episode from season 1.