Q. Why did Hashem pick the 10th of Tishrei as Yom Kippur?
A. After the sin of the golden calf, Moshe ascended to Mount Sinai to receive the second set of commandments. For forty days, Moshe and Bnai Israel were praying and fasting. At the end of the forty day period, on the 10th of Tishrei, Hashem said, that He has pardoned their sins and gave Moshe a new set of tablets.
On Yom Kippur, man is pardoned by Hashem; however, not by his fellow man, thereby, he must go to the people he has forsaken and ask for their forgiveness.
Moreover, Hashem does not forgive those who are lax about observing the Torah throughout the year because they think that they just can do teshuva-repent- on Yom Kippur and be forgiven.