There is one supreme reason you should care about people in far away places: Because the Lord commanded us specifically to do so.
The last words He spoke before ascending into heaven include this commission to each of us:
Go and make disciples of all nations.
Mathew 28:19
He said “Go”, which is the opposite of staying to reach just our own. He said “make disciples”, which is a long process requiring missionaries to live and do ministry among the people. He said “all nations” which means many nations other than our own, and other than those which are easy to reach or fun to visit.