"Three mitzvoth were the Jewish people commanded when they went up [to Jerusalem] on the festival, re'iyah, chagigah and simcha"(Chagigah 6b). These three obligations of "seeing, holiday and joy" refer to the threefold obligation of sacrifices that one was to bring on the shalosh regalim.
When coming to visit someone's house, especially if one plans to stay a while it is to be expected that one bring a gift for the host. The gift is meant as a token of appreciation and whether the host actually needs it is of secondary importance. We bring the olat re'iyahsacrifice to express our thanks for having the opportunity to dwell in G-d's home. As a gift we do not partake of it ourselves - with all being offered on the altar.
"There is no joy except with meat and wine" (Pesachim 109b). This teaching was said first and foremost regarding the korban chagigahthe festival offering which was to be eaten on Yom Tov. It served for example as the "main course" at the Pesach seder with the pascal lamb serving as desert.
In the biblical way of thinking meat was ideally meant to be eaten in the context of sacrifices, thereby elevating the act of eating - especially when it involves the spilling of blood - into something more spiritual. We were to offer parts on the altar, share some with our religious leaders (kohanim) and enjoy the rest with our family and friends. All the while there was only true joy if one ensured "the stranger, the widow and the orphan" could also partake of the festive meal. As sacrifices had to be consumed within a day or two there was plenty of opportunity (and need) to share with others.
In addition to the re'iyah and the chagigah a third category of sacrifices were brought "the shlamei simcha, the joyous peace offering which complemented the chagigah.
While the Torah mandates that we bring these korbanot on Yom Tov the amounts spent on each korban or even exactly which animal to bring was left to the individual. How one should allocate one's resources between the olat re'iyah where all is sacrificed to G-d and the shelamim where most is eaten by man is a debate between Beit Shammai and Beit Hillel. The school of Shammai ruled that we should spend two maot on the olah and only one on the shelamim. Beit Hillel suggested the opposite.
This debate is just one manifestation of the differing worldviews of the houses of Shammai and Hillel. The Gemara (Chagigah 12a) records a fascinating debate between them regarding the order of Creation. Beit Shammai suggested that G-d first created heaven and then earth. Beit Hillel argued the opposite suggesting that the earth was created first [1]. This strange debate seemingly has little to do with the order of creation something no human can ever know. But it has much to do with the purpose of Creation. Should our primary concern be heaven or earth; G-d or man; Judaism or Jews? It is no accident that Beit Shammai are stricter in their application of Jewish law and Beit Hillel more lenient. There is little room for compromise when focusing on fulfilling G-d's will but much need for it when responding to man's needs. It is so difficult to fully live up to G-d's demands that Shammai argued it would have been better had man not been created in the first place (Eiruvin 13b).
In a well-known story the Gemara relates how a potential convert asked to be taught the entire Torah while standing on one foot. Shammai most understandably threw him out of the house - this ridiculous request turns G-d's Torah into little more than a sound bite. Hillel on the other hand focused on bringing people under the loving canopy of G-d - slowly teaching the depths and beauty of Torah.
[1] While the first verse of the Bible would seem to indicate that the heavens were created first in the second account of creation in chapter two of Breisheet the creation of the earth is mentioned first. This is just one of many "contradictions" between these two biblical accounts. For one approach to explaining such see Rabbi Soloveitchik's Lonely Man of Faith.
