Why did Kennedy think that ending the Cold War would benefit both sides and why did he think relations between both countries could change?

If we follow the speech given at American University in May of 1963, along with some of the policies he began to initiate before his death, we can see that Kennedy articulated the idea that ending the Cold War is something that will make "life on Earth worth living."  Through his experience with the Cuban Missile Crisis, Kennedy was brought close enough to the brink of nuclear war 90 miles off of the American coast.  These 14 days brought American and Russian conflict to a dangerously close level, and many believe that such an experience casted a major shadow on Kennedy, leading him to embrace the idea that ending the Cold War was the only way to avert another such incident.  In his commencement speech at the American University, Kennedy argued that rational individuals must embrace a perspective of peace and ending the Cold War:

Today the expenditure of billions of dollars every year on weapons acquired for the purpose of making sure we never need them is essential to the keeping of peace. But surely the acquisition of such idle stockpiles -- which can only destroy and never create -- is not the only, much less the most efficient, means of assuring peace. I speak of peace, therefore, as the necessary, rational end of rational men. I realize the pursuit of peace is not as dramatic as the pursuit of war, and frequently the words of the pursuers fall on deaf ears. But we have no more urgent task.

In making the economic argument that the escalation of the Cold War will cause incredible hardship on the national budget, Kennedy was ahead of his time.  The call to end nuclear escalation was buttressed with the call to reexamine national attitudes towards the Soviet Union and embrace the paradigm of incremental change where peace is something that was to be realized "in our lifetime."  The most famous of this hopeful vision of American foreign policy was revealed in its conclusion:  "And if we cannot end now our differences, at least we can help make the world safe for diversity. For in the final analysis, our most basic common link is that we all inhabit this small planet. We all breathe the same air. We all cherish our children's futures. And we are all mortal."