The mishna in Taanis says that Tu B'Av and Yom Kippur were the 2 happiest days of the year. Tu B'av for a number of reasons - mostly to do with marriage: the girls would go out dressed in white and offer themselves for marriage, also that is the day the shevatim were allowed to intermarry (between shevatim), among other reasons.
Regarding Yom Kippur it brings a passuk that says "Daughters of Israel go out with the crown on the day of the wedding and the day of happiness." "The day of the wedding" refers to Yom Kippur, as that is the day Hashem married the Jews (because of mattan Torah taking place on Yom Kippur) and the day of happiness because on Yom Kippur our sins were forgiven (by receiving the second Luchos on Yom Kippur, Hashem showed us that he had forgiven us for the golden calf).
This is a very interesting juxtaposition between Tu B'Av which is obviously a happy day and Yom Kippur, on which one rarely sees a smile on anybody's face and can hardly be thought of as being described as a happy day. But the happiness is similar and they are both based on the ideas of marriage. on Tu B'Av we are happy because of marriage and on Yom Kippur because of a metaphorical marriage.
Another point is that on the day of one's wedding, it is brought down in the seforim that all of his sins are forgiven and he fasts (similar to Yom Kippur).
I think the connection between these different days is this idea. Marriage is an opportunity. It is similar to the opportunity we are given on Yom Kippur. On Yom Kippur we repent and we are given a new start with a clean slate. On Yom Kippur, the day we start with a clean slate, we were given the Torah. The Torah is the users manual for a Jew's life. It tells us how to live and how to act. When you are starting anew, you should be coming without any pre-conceptions and pre-existing ideas of how things should be done. You must read the users manual for instructions how to live properly.
Marriage is the same. You are being given a fresh start. Your past is behind you and your future is what lies ahead of you. Until now you were a single guy without a care in the world. You could wake up late, you could hang out all day, you could be productive or not. It affected mostly just yourself. Now you have someone else in your immediate circle. You are starting anew, with your wife. Everything you do now will affect not just you, but her. When you are considering doing something, you have to think about it differently than you did previously. You have to consider if it is appropriate and proper for your wife as well.
The new beginning, presenting you with new opportunities, is what effects the great simcha.