When those that are close to us die, it affects us because we no longer have that influence in our life. The closer that individual is to us, the more noticeable the influence that they have upon us. Of course when it is only an acquaintance, the influence they have upon us often is seemingly simple and small. Of course we don't keep in mind that most acquaintances we have are through those individuals that we know. So our acquaintances may not have a large influence upon us, but they may heavily influence those that are close to us and those that are close to us have a heavy influence upon who we are. Of course the bigger stretch is for those that we do not know. However, undeniably through a similar chain of interactions, those individuals we do not know do influence us through the interactions they have with others that in turn influence others until that chain of influence filters down to us. Donne communicates this through sharing the idea that "no man is an island," for we are all a part of the continent that is the world of human existence.
Donne uses the ideas of the bell tolling and the influences that we have upon each other to communicate that when someone dies a bit of us dies as well. He warns us to realize that the bell actually tolls for us when it tolls for another because when we lose any part of the whole that is the human family, we lose a part of ourselves. The influences along the chain that eventually make their way to us influence, at least a tiny bit, who we are. So it really is interesting to think that we influence each other in such ways that every action or lack of action truly affects the whole of mankind.
This helps to illustrate even more the importance of being our best selves always. When we interact with others, that influence continues down an infinite line that continuously influences the whole of the human family. That influence we have upon others, no matter how small, has at least some part in influencing all of mankind. Undeniably that influence is almost imperceptible, but when we have such opportunities to make a difference, how can we not choose the better part? Knowing how this chain of influence continues down throughout time, it is so crucially important that we develop good relationships with others and do everything we can to help them become great people so that they can in turn share that with others.