For most of human history, the birth rate and death rate have been pretty well matched, in large part due to limited resources. As more resources became available, a core population would be able to grow. If stress became too great, some people would move, taking their chances locating other resources, and in the process relieving the stress on those who remained. Over time, successful splinter groups might trade resources with other groups, effectively increasing the resource base of everyone.
Having taken over our entire planet, humanity is once again resource limited, and to survive our birth and death rates must once again approach the same value. Space exploration offers the same benefits to our “core population” that other regions of Earth offered our ancestors: Relieving stress by splintering off part of the population, with possible access to other resources. This is in addition to improving the chances that our species will survive if some catastrophe makes Earth uninhabitable. Fortunately, we have the extra resources to enable this “calving” of humanity, but we can’t count on those resources being available much longer.