What does Pope Francis mean by a big heart? A pastor for whom I once worked often preached on this at weddings. Authentic sacramental love is self-giving, and ever-widening. It extends to friends and neighbors in need, and even to the poor and needy. Such additions to married love inspire others to notice and to remark, “This is good. This is of God.”:
196. In addition to the small circle of the couple and their children, there is the larger family, which cannot be overlooked. Indeed, “the love between husband and wife and, in a derivative and broader way, the love between members of the same family – between parents and children, brothers and sisters and relatives and members of the household – is given life and sustenance by an unceasing inner dynamism leading the family to ever deeper and more intense communion, which is the foundation and soul of the community of marriage and the family”.(Familiaris Consortio 18) Friends and other families are part of this larger family, as well as communities of families who support one another in their difficulties, their social commitments and their faith.
St John Paul refers to a “dynamism” in a family that looks outside of its four walls of home. I would agree. In our world in which so many people are adrift from the nearness of extended family, such associations are especially important. Check Amoris Laetitia online here.
Catholic Sensibility 
I think of a large-hearted or big-hearted person as someone who is fundamentally open and disposted to trusting as a default (though circumstance and experience may cause them to distrust specifically).