Showing that we can afford space exploration, globally speaking, is not nearly the same as showing why we should pay for it. Despite the fact that we can make room for it in our list of priorities if we choose to, there are still many other priorities that are suffering from a lack of (or those that have died from an absence of) funding. Given budgetary realities, why should we fund space exploration?
Space exploration has brought many direct benefits in the years since its inception, though perhaps none that most people recognize that we could not live without. One of the most prominent successes is the Hubble Space Telescope 
(and other space-based observatories, as well), without which any number of insights into the workings of the cosmos may not have been made.
Scientific knowledge is useful, but most citizens measure the effectiveness of a program relative to its positive or negative impact upon their lives. To increase public awareness of the benefits of space exploration, NASA has gone so far as to establish a website dedicated to elaborating on some of the successful technology transfers (http://techtran.msfc.nasa.gov/at_home.html) between the American space program and everyday life; perhaps more significantly, the website also mentions their dedication to looking for "spin-in" opportunities, as well. Borrowing ideas and technologies from outside the space sector stimulates the economy by providing a market for those technologies, provides incentive for small businesses and researchers to form partnerships with the public sector, and reduces the development costs to taxpayers.
Some of the more significant spin-offs include improved firefighter breathing gear, rescue jaws, scratch-resistant lenses, and spacesuit fabric (used as roofs for sports stadiums, malls, and airports). The variety of spin-offs makes it clear that the connections are not always direct or obvious- a case in point is the use of image processing techniques that calculate the height of terrain to analyze the effectiveness of makeup! The Spinoff magazine (http://www.sti.nasa.gov/tto/spinfaq.htm) NASA produces is dedicated to pointing out connections between the space program and business, though some of these cases are a matter of NASA lending expertise rather than transferring technology specifically developed for space uses. (Contrary to popular belief, Tang, Teflon, and Velcro are NOT spinoffs of developments in the American space program.) Despite the effect of the Constellation Project on other aspects of NASA's research budget, it still plays a significant role in US research and development.
Not all the benefits of space exploration are direct- for instance, some of the engineering experience acquired can be put to use on other projects incorporating similar technologies or operating in extreme environments such as Antarctica and the deep ocean, to name two prominent examples, and managers with experience in coordinating high technology projects are a valuable national or, in this increasingly globalized world, international asset. Both China and India recognize their space programs as a critical component of STEM (science, technology, engineering, and mathematics) education, since such an effort serves as a powerful draw and a source of jobs for talented individuals- it is also no coincidence that it may also help to reduce the rate of high-tech "brain drain" towards Western countries. [ad name="Adsense Small Horz Banner"] Space exploration can also serve as a source of national pride or an economic asset; one need only look so far as the space race that occurred between the United States and Soviet Union for an example of the former or the European Space Agency's Ariane rocket for an example of the latter. Indeed, in the early twenty-first century, access to space in the form of communications, navigation, and earth observation assets seem almost essential to an extremely high standard of living. High-bandwidth satellite communications and transportation efficiency make possible just-in-time inventories, Dish TV, and direct point-to-point aircraft flights.
From a scientific perspective, a number of revelations have occurred specifically due to our active exploration of outer space; take, for instance, the fact that we now know with a large degree of certainty that there was water present on Mars' surface at some point in the past, a revelation that we would have been unlikely to arrive at by simply observing Mars from a distance. That better understanding of Mars gives us an appreciation for and knowledge of how the scientific laws and theories we've developed here on Earth apply elsewhere in the universe- it adds a context to that knowledge, just as a knowledge of geology, meteorology, climatology, and chemistry provide context to biology. And, from a philosophical perspective, long-term space exploration is likely to challenge our sense of self and place, our decision-making processes and organizational structures, and our ability to construct "built environments" and ecologies (or perhaps arcologies) that can sustain us and allow us to grow in a multitude of dimensions, physical and otherwise.
Considering all of the above, it would seem that there is a direct benefit from much of the unmanned space exploration that is being carried out; in terms of maybe the correct question to consider is, "what does manned space exploration do for us?" I think that, a fundamental way and like other explorations that have come before, it prepares us to make an informed decision about what we want to become. To this point, Earth has defined us as it has all living things we share this planet with; as we move outward, we may have no choice but to move inward, changing who we are in such a significant way that, while we will still (and always) be Earth's children, we will no longer be defined by her. In the long run, this may well prove to be our own Copernican revolution.