If you haven't read it yet, see the post on Tamuz, as much of it also relates to the month of Av.
Mishenichnas Adar marbim b'simcha; mishinichnas Av mimatin b'simcha
The Chozeh of Lublin explains these words in a fascinating manner: When the month of Adar arrives, you should strengthen the force of kedusha with simcha, and when the month of Av comes, you should lessen the power of klipos and the sitra acher with simcha. This would seem congruous with the teaching that not being b'simcha increases the power of the sitra acher and that being in a depressed state can increase the power of the sitra acher.
R' Shimon Shwab used to say that by the month of Av the mishna says that we should lessen our simcha, not to totally get rid of it; one always needs some level of b'simcha.
Kibud Av & Tikun Chatzos
The month is called Av, which means father. We have to also have Kibud Av for our Father in Heaven. He has no home, so to speak; the Beis Hamikdash is still not rebuilt. We need to show in some way that we are concerned for our Father. One way to do that is with Tikun Chatzos. Many people say it at midday, during the "three weeks", in addition to saying it at midnight. In the Yeshiva of the Chasam Sofer, the Yeshiva said it together. If one isn't able to say the whole thing, then they can just say the chapters of tehilim. Even if one says just a few chapters of tehilim from Tikun Chatzos, it shows that we are at least a little concerned about the churban.
Hoping for the Geulah
There are 12 different permutations of Yud-Kei-Vav-Kei. Minhag Yisroel is to say the permutation הוי״ה when referring to Hashem's Name, since it is forbidden to say it in its proper order. Why do we read it in this order as opposed to the others? Because Yud-Kei-Vav-Kei written in the proper order is that of Nisan, the month when we were redeemed from Mirtzrayim, similarly now, we use this permutation of Av, the month when Hashem will take us out of golus in the future. By doing this, we show our hope that we will be redeemed just as we were from Mitzrayim. (Bnei Yisaschor, see there for much more in detail)
Why the Satan Doesn't Want Us to Make a Siyum
There is a minhag by Talmidei Baal Shem Tov to make a siyum on a Mesechta of Gemara, specifically during the 9 days. The reason for this can't be that they are simply interested in eating meat, because the rest of the year these same people stay far away from any material indulgence. The reason they make the siyum is because the Seforim HaKedoshim say that a seuda for a siyum on a Mesechta weakens the power of the ס״ם, and "he" does not want it to be done. This can be seen from a fascinating allusion; the letters of סמא׳ל are the first letters of ס׳יום מ׳סכת א׳ין ל׳עשות (Siyum Mesechta don't do). He is the representative of Esav, who has control over these months, so we purposely want to make a siyum then. (Shaar Yisaschar)
עַל-כֵּן לֹא-יֹאכְלוּ בְנֵי-יִשְׂרָאֵל אֶת-גִּיד הַנָּשֶׁה The Zohar says that eating on Tish B'Av is like eating from the gid hanoshe. The word אֶתstands for the words ת׳שעה א׳ב. You can now read the posuk as "Therefore the children of
Similarly, the words אֶת-גִּיד הַנָּשֶׁה, are the same gematria as תשעה אב. (Imrei Pinchos)
This all fits in very well. The Malach of Esav was the one that injured Yaakov Avinu and therefore we can't eat from the gid hanoshe. This month of Av belongs to Esav and it is when the Beis Hamikdash was destroyed. That, on a simple level can help us explain why the Zohar compares eating from the gid hanoshe and eating on Tisha B'Av; they are both giving power to Esav.
Further, the Malach of Esav, who fought with Yaakov Avinu told him to only call himself Yisroel from now on, not Yaakov. His intention was that he shouldn't use both names becuse the gematria of יעקוב and ישראל together (Yaakov is sometimes written with a "vav") is the same as שטן קרע, which is a name with the power to eliminate bad forces, chitzonim. (Toras Moshe-Chasma Sofer)