Quick approximate summary:

• John von Neumann first says, essentially, that the Industrial Revolution has made the world smaller, and that in earlier centuries the problems caused by it were contained to nations, but that now they extend to the entire globe.
• For a first example, he talks about cheap energy and industrial synthesis, though he predicts that nuclear fusion and transmutation would be much more practically feasible than they have turned out to be.
• He briefly mentions expected major improvement in automation, communication, and transportation.
• He then talks about anthropogenic climate change and the broad possibilities of geoengineering the climate.
• All of these technologies can vastly improve human life, or destroy it.
• He immediately rules out the "solution" of [preventing advances in technology] as both undesirable (because it blocks the positive uses) and impossible (because it would require total coordination and a total change in human values).
• He next considers the possibility of permanently avoiding war through diplomacy etc, and does not think that the 1950s drive for world peace will last long; furthermore, such an initiative would need to adapt to ever-more-powerful technologies as fast as they are introduced.
• He frames the upcoming decades as a dangerous but useful evolution, where we will either succeed or fail catastrophically, and doesn't sound especially optimistic. Our best hope is to innovate new political forms that are capable of handling major threats with patience, flexibility, and intelligence.