In this quote, Hamlet is contemplating death. He is a pretty depressed fellow, and finds everything in his life "vile" and "rank." So, because life is so miserable, he wonders to himself why he, and by extension everyone, won't just die to escape the world. What keeps people from leaving this life to go on to whatever happens next? This is all pondered in his famous "To be or not to be, that is the question" soliloquoy, which is so well-known across the world (well, at least the first few lines are).
The quote above refers to death--to Hamlet, death is an "undiscovered country." It lies on the other side of life, it is mysterious, and one's soul has yet to discover it. He uses a metaphor to compare what lies in store for us after dying to a country that we will, after we die, explore. Thus far, death is still a mystery to mankind, or, undiscovered. We do not know what lies after death. We have not discovered this for ourselves, and been able to proclaim it to all mankind. That is why death is an "undiscovered country." And, those that go to that country, or who die, are travellers whose bourne (or journey there) is never returned from. People who die usually don't come back to tell us all about it. They are travellers in that country of death, and they never return once they go there. Does that explanation help to clarify the quote for you a bit?
Psychology comes into play because death is a puzzling and frightening thing for all people. It is something that is a huge part of our psyche. Dealing with death is an awful, terrifying thing that all people must cope with at some point in their lives. So, death becomes a necessary part of our mental health; how we deal with it, what we believe about it, and how we approach it when we encounter it. The philosophical, religious, spiritual, and emotional discussions and implications of death are limitless. Psychologists spend a lot of time dealing with the issue with clients, and it is a necessary fact of life that we all must cope with.
I hope those thoughts helped; good luck!