The Quran

لِيَشهَدوا مَنافِعَ لَهُم وَيَذكُرُوا اسمَ اللَّهِ في أَيّامٍ مَعلوماتٍ عَلىٰ ما رَزَقَهُم مِن بَهيمَةِ الأَنعامِ ۖ فَكُلوا مِنها وَأَطعِمُوا البائِسَ الفَقيرَ

Muhammad Asad

so that they might experience much that shall be of benefit to them,¹ and that they might extol the name of God on the days appointed [for sacrifice], over whatever heads of cattle He may have provided for them² [to this end]: eat, then, thereof, and feed the unfortunate poor.³

Lit., "that they might witness benefits [accruing] to them" - i.e., increased consciousness of God through facing the first temple ever dedicated to Him, as well as the consciousness of being part of a brotherhood embracing all believers. Apart from these spiritual benefits, the annual Pilgrimage to Mecca provides an opportunity for believers from all parts of the world to become acquainted with the many social and political problems that confront the various geographically separated sectors of the community.
The repeated Qur'anic insistence on pronouncing the name of God whenever one slaughters an animal is meant to make the believers "realize the awfulness of taking life, and the solemn nature of the trust which God has conferred upon them in the permission to eat the flesh of animals" (Marmaduke Pickthall, The Meaning of the Glorious Koran, London 1930, p.342, footnote 2). As regards the "days appointed" (ayyam ma'lumat) spoken of above, they apparently denote the Festival of Sacrifices, which falls on the 10th of the lunar month of Dhu'l-Hijjah, as well as the next two days, marking the end of the pilgrimage (Ibn 'Abbas, as quoted by Razi).
Whereas the pilgrims are merely permitted to eat some of the flesh of the animals which the, have sacrificed, the feeding of the poor is mandatory (Tabari and Zamakhshari) and constitutes thus, the primary objective of these sacrifices. Apart from this, they are meant to commemorate Abraham's readiness to sacrifice his first-born son after he dreamt that God demanded of him this supreme sacrifice (see 37:102-107 and the corresponding notes); furthermore, they are a reminder that God is the Provider of all sustenance and the One who gives life and deals death, and that all must return to Him; and, lastly (as stressed by Razi), they are to be symbols of each believer's readiness in sacrifice himself in the cause of truth.