Background of Torah reading:
Moshe Rabbeinu established for the Jewish people that on every Yom Tov one is to read the Torah portion dealing with that Holiday. The Sages of the Mishneh and Gemara chose the exact portion of the Torah that is read on each holiday.
Haftorah: The reading of the Haftorah was instituted by the Anshei Kneses Hagedola. They instituted that one should read from Navi on every holiday from a portion that deals with the holiday events. The Sages of the Mishneh and Gemara chose the exact portion of Navi that is read on each holiday.
The Maftir: [Maftir in truth refers to the reading of the portion from Navi formally known as the Haftorah. It comes from the word Petor which means finish, as the reading is completed with the Haftorah.] However, the person that reads the Maftir from Navi is also required to first read a section from the Torah. In the times of the Sages of the Mishneh and Talmud the Maftir would read from the Torah as part of the five Aliyos that were read from the portion of that Holiday which was read from the first Torah scroll. [There was hence no second scroll taken out from the ark] However, in later generations, the Rabanan Savuraiy established, and so was accustomed by the Geonim which followed them, that the Maftir is to read [from a second scroll] the portion of the Musaf sacrifice written in Parshas Pinchas. This institution was based on the following teaching of the Sages: Avraham stated in front of Hashem “Master of the world when the Temple is not in existence the children of Israel do not have on what to find support. What will be with my children when they sin? Perhaps you will do to them as you did to the generation of the Mabul and Haflaga. Hashem answered “I have already established for them the order of the Karbanos. When the Karbanos are read, I consider it as if they sacrificed the Karban before me and I forgive all of their sins. [Regarding why the Maftir is read as the sixth Aliyah and is not part of the five Aliyos-see chapter 282/10-13!]
Two Torah scrolls: Being that two different sections of the Torah need to be read on Yom Tov, therefore, one is initially required to remove two Torah scrolls; one for the Holiday reading, and the second for the Maftir. It does not suffice to remove only one Torah scroll and then roll it to the Parsha of Musafim [for Maftir].
 Admur ibid; 282/1; M”A 135/1; 535; Rambam Tefila 13/8; Mishneh Megillah 31a
The hint in scripture: The above institution is hinted to in the verse “Vayidaber Moshe Es Moadeiy Hashem El Bnei Yisrael”, from which we learn that Moshe established that each Prasha of the Moadim is to be read at its proper time. [Admur ibid]
 Admur ibid; Megillah ibid
 Admur 282/1; 488/5; M”A 135/1
 The reason: The reason behind the different number of Aliyos [Three during the week; Four on Rosh Chodesh; Five on Yom Tov; six on Rosh Hashanah; seven on Shabbos] is based on the levels of holiness that each day contains over the other days. See 282/1 for further details on this matter.
 Admur 488/5
 Admur ibid; Sefer Hamachria 31 in name of Rabbeinu Tma that it was instituted by Ezra
 Admur 488/6
 Admur 282/10; 488/6
 Admur 488/6; Michaber 144/3
 One must read from the Parsha of the Holiday, as per the institution of Moshe Rabbeinu, and one must also read the Karbanaos from Parshas Pinchas, as per the instititution of the Rabanan Savuraiy.
 The reason: As one may not roll a Sefer Torah in front of the congregation due to respect of the congregation. [Admnur ibid]