(Just so you get why we extend the feelings of Tisha B’Av back to the start of Av.)
Sometimes the nature of a feast day or fast day ends up defining the entire month in which it occurs.
משנכנס אדר מרבים בשמחה: When Adar enters, we increase in Joy.
משנכנס אב ממעטים בשמחה. When Av enters, we decrease in joy.
Pesach plays a role in giving the entire month of Nissan a festive character, in that we don’t give eulogies or fast or say certain sadness-inducing prayers the whole month.
Why are those months singled out?
I have a theory that it is the sources in Tanach that tell us to treat the whole month as special.
“Shamor es Chodesh HaAviv”. Observe the Month of Aviv. I.e. Pesach is observed (in a way) during the whole month, even though Pesach is only one week long.
|כ וַיִּכְתֹּב מָרְדֳּכַי, אֶת-הַדְּבָרִים הָאֵלֶּה; וַיִּשְׁלַח סְפָרִים אֶל-כָּל-הַיְּהוּדִים, אֲשֶׁר בְּכָל-מְדִינוֹת הַמֶּלֶךְ אֲחַשְׁוֵרוֹשׁ–הַקְּרוֹבִים, וְהָרְחוֹקִים.||20 And Mordecai wrote these things, and sent letters unto all the Jews that were in all the provinces of the king Ahasuerus, both nigh and far,|
|כא לְקַיֵּם, עֲלֵיהֶם–לִהְיוֹת עֹשִׂים אֵת יוֹם אַרְבָּעָה עָשָׂר לְחֹדֶשׁ אֲדָר, וְאֵת יוֹם-חֲמִשָּׁה עָשָׂר בּוֹ: בְּכָל-שָׁנָה, וְשָׁנָה.||21 to enjoin them that they should keep the fourteenth day of the month Adar, and the fifteenth day of the same, yearly,|
|כב כַּיָּמִים, אֲשֶׁר-נָחוּ בָהֶם הַיְּהוּדִים מֵאֹיְבֵיהֶם, וְהַחֹדֶשׁ אֲשֶׁר נֶהְפַּךְ לָהֶם מִיָּגוֹן לְשִׂמְחָה, וּמֵאֵבֶל לְיוֹם טוֹב; לַעֲשׂוֹת אוֹתָם, יְמֵי מִשְׁתֶּה וְשִׂמְחָה, וּמִשְׁלֹחַ מָנוֹת אִישׁ לְרֵעֵהוּ, וּמַתָּנוֹת לָאֶבְיֹנִים.||22 the days wherein the Jews had rest from their enemies, and the month which was turned unto them from sorrow to gladness, and from mourning into a good day; that they should make them days of feasting and gladness, and of sending portions one to another, and gifts to the poor.
So the Megillah clearly says that all of Adar has this Joy.
And then there is Tisha B’Av. One thing you may not have realized is that nowhere in Tanach is there a verse that says the Beis Hamikdash was destroyed on the 9th of Av. Both in Melachim (II Kings) and Yirmiahu (Jeremiah), the month is the time it was destroyed.
Now while the entire months of Adar and Nissan are imbued with this Simcha, we can thank someone in the Jewish Calendar department for not pasting Tisha B’Av’s attitude onto the entire month. But it does start with Rosh Chodesh, and extends to the day after Tisha B’av.
I think that is the deeper meaning of the Hagada’s phrase “Yachol Mei’Rosh Chodesh”. One can reasonably assume that the presence of a notable day in a month can be felt at the start of that month, due to the nature of anticipation; and can spread throughout the whole month.