Ín the Gemara, Pesachim 115b, Shmuel says the matzah is called לחם עוני (lechem oni) because we say a lot of things on this matzah.
Ín truth we say the Haggadah and Hallel on everything on the ka’arah, not just the matzah. So, why is the recital associated specifically with the matzah?
The Kozhnitzer Maggid says that matzah contains 3 of the 4 foundational elements – earth, fire and water. The air is missing, because the matzah has to be made so quickly. The Kedushas HaLevi connects the four foundational elements of earth, air, fire and water to the four letters of Hashem’s name. We also see 3 of the letters of Hashem’s name in the word matzah, using the code of את“בש (reversing the letters so Alef = Tav, Beis = Shin, Gimmel = Reish, and so on).
מ becomes י
צ becomes ה
And the final ה of Hashem’s name is already in the word מצה.
Only the vov is missing. Ín the Zohar it is brought down that the letter vov represents the connection between above and below. The vov corresponds to the voice (air). When the Yidden brought Bikkurim, the possuk says: ואמרת וענית .Rashi said that “v’anisa” means it needs to be done out loud, so now we understand why there is a minhag to say the Haggadah out loud. When we say the Haggadah out loud, we add the vov of Hashem’s name to the matzah.