The creed agreed at Constantinople in 381 says that the Holy Spirit proceeds from the Father. Latin-speaking churches came to recite it with two words added, filioque, and the Son.
The phrase appears first in Spain, at a council of Toledo in 589 held to receive the Visigothic kings out of Arianism, where affirming that the Spirit proceeds from the Son too helped assert the Son's full divinity. It spread through the Frankish church and was promoted by Charlemagne's court, which used it polemically against Byzantium. Rome resisted for two centuries: Pope Leo III accepted the theology while refusing to alter the creed, and reportedly had the original text engraved on silver shields at St Peter's. The addition was received at Rome only in the eleventh century, at which point it became the Western norm.
The Eastern objection has two distinct components, and conflating them obscures the dispute.
The procedural objection is that a creed promulgated by an ecumenical council can be changed only by another ecumenical council, and that a portion of the church acting alone has no such competence. On this reading the offence is not the word but the unilateralism, and it is the sharper of the two objections because it concerns who holds authority in the church.
The theological objection is that within the Trinity the Father alone is the source, the sole principle from which the Son is begotten and the Spirit proceeds. Photius, patriarch of Constantinople in the ninth century, argued that a double procession either introduces two first principles or collapses the distinction between Father and Son. Western theology answers with Augustine's account of the Spirit as the mutual love of Father and Son, and with the formula that the Spirit proceeds from the Father through the Son, which the East has historically found acceptable.