The main context to Kennedy's 1960 Inaugural address is the Cold War. The address, therefore, is aimed not just at the United States' people, but those of the entire world. One of its key goals is to allude to the main differences that the United States wanted to articulate as separating it and its allies from the Soviet sphere of influence.
Therefore, Kennedy opens his speech mentioning "freedom," a clear allusion to the lack of freedom in the communist states. Kennedy also very quickly mentions God, asserting that he swore his oath of office before "Almighty God" and stating that our rights come not from the "state but the hand of God." This clearly marks a line of demarcation between the "godless communists" and the "god-fearing" democracies of the world. He will also end the speech by evoking God.
Kennedy extends a hand across the globe, asserting American leadership of the free world. Kennedy assures allies of his fullest support and reaches out to poorer parts of the globe, such as the Global South. At this time, there was a contest for which sphere of influence many of these former colonies would fall under—American or Soviet—and Kennedy appeals to these undecided nations on the basis of freedom.
Kennedy also asserts strongly that the United States will not back down in defending its interests or those of its allies. He asks for cooperation and reconciliation, not naming the Soviet Union but clearly intending that nation to understand his message of being willing to use both military might and diplomacy in the Cold War.
This address is important because it strongly asserts the United States' position as the "leader of the free world": the days of isolationism are long gone, and the United States will do everything it can to promote its way of life around the globe.