A favorite Rabbi of mine, observing the trend of increased attendance, and bemoaning the lack of similar participation throughout the rest of the year, once quipped that Yom Kippur is the one day in the Jewish calendar on which it is not necessary for a Jew to be in shul. You see, Yom Kippur is not merely a day of communal supplication; the day's passing itself atones for an individual's sins. One simply needs to adequately prepare and make sure they do not interfere with this process, which otherwise runs of its own accord. So, while many believe that attendance in shul on this one day is mandatory, being optional throughout the rest of the year, the opposite is true. A Jew should come to shul every day to elicit their daily portion of spiritual and physical sustenance, which is dependent on their daily supplication. On Yom Kippur, however, the one day on which everyone runs to shul, this isn't strictly necessary because the day itself atones.
None of this is to dilute the importance of communal prayer and supplication, including on Yom Kippur, but even more vitally every other day of the year. It is merely a recognition that on Yom Kippur, a single tear or thought of repentance is sufficient for the day to atone, while throughout the rest of the year, atonement requires greater individual struggle, and benefits from communal effort.
I was long fond of this narrative, but have become more appreciative of the desire of communal supplication on Yom Kippur. Reading through the day's liturgy, one is liable to notice that many of the supplications are written in the plural "we" form. Indeed, the standard daily prayer of supplication - the Tachanun - employs "we", as in, "We have transgressed, we have acted perfidiously, we have robbed, we have slandered", etc.
Why "we", and not "I"? There are various explanations. One is that the supplication on Yom Kippur is loudly vocalized, and we do not wish anyone to be embarrassed who is atoning for the specific sins in question. If one person is chanting, individually, "I have robbed, I have slandered", the rest of the community might reasonably demand to find out who it was they robbed and slandered. I should point out that part of an individual's repentance is to rectify their wrongs, in our example, returning what they robbed (with interest) and discrediting the slander. Once that is performed, their public penitence, confessed in plural form so as not to draw unnecessary attention, is accepted.
Another explanation, popular today among "tikkun olam" progressives, is that no individual act of transgression is performed in a vacuum. Indeed, every action an individual takes can be attributed, at least in part, to societal conditioning and socio-economic or political circumstances beyond the individual's control. Therefore, the community must share responsibility for the actions of its members, including atoning for them. Complementary to this notion is the idea that there are certain decisions which a community makes together and must atone for together - social and foreign policies are favorite examples. All these reasons necessitate the use of the plural pronoun "we" in communal supplications.
Never an admirer of collectivism, which often trends toward coercion, I prefer a more individualistic explanation. Communal atonement is about recognizing that no one individual's repentance is complete without all our individual repentance being complete. It's almost as if we're standing with each individual as they complete their particular obstacle course. Only through their individual effort will the task be completed, but the community stands at the ready, providing strength and encouragement. As each successive individual completes the obstacle course and is purified of their transgressions, we reestablish the perfect unity among the souls of Israel, which emanate from one root. So purified and unified, we can present ourselves to the heavenly court, certain of a decree for good.
It is true that communal attendance at a Yom Kippur service isn't strictly necessary, and it is certainly no substitute for praying at shul on a daily basis. However, there is something quite special and redeeming about the desire of Jews, largely unaffiliated through the year, to stand up and be counted among the Jewish people on this one day, to accept the community's support as they progress through their own obstacle course of repentance, and to help others in kind.