If by liturgy we mean a service with a fixed structure and set forms of prayers then the answer is that it is descended from the Church's worship in the earliest days. Some prayers (e.g. the Lord's Prayer) are of course found in the Bible or are based on the prayers of the synagogue (e.g. the prayers "Blessed are you..." said over the bread and wine). The various Eucharistic Prayers follow a pattern first mentioned in about 150AD. The chant "Lord, have mercy" (Kyrie eleison - the fact that we still sing it in Greek shows just how ancient this prayer is) and the Creed similarly date back to these early centuries.
By using these texts, we are witnessing to our fellowship with Christians all through the ages. Similarly, by using words and structures that are being used by other worshippers today, we are taking part in something that is larger than the worship offered by one group of people in one building at one time.
Over the centuries, local variations crept into the liturgy. Some of these were then adopted by churches in other places. The words invoking the Holy Spirit to consecrate the bread and wine originated in Spain in the early Middle Ages and were then taken over by the Church as a whole. Some prayers moved about over the years, e.g. the hymn "Glory to God in the highest" was moved to the end of the service at the Reformation but back to its original place near the beginning in most modern versions.
After King Henry VIII broke away from Rome the liturgy was simplified and under his son, Edward VI, a new prayer book was produced, now in English for the first time. It was called the Book of Common Prayer and reached its definitive form in 1662. The word "common" means "the same for everyone, everywhere". This was designed to unite the Church in its form of worship, so that everyone would feel at home with the service, no matter where they were. It also promoted a shared belief - people could not take liberties, making up prayers that reflected ideas that were not part of the Church's teaching.
Since Henry VIII's day, the Anglican Church has spread all over the world. It has produced local prayer books and, back in England, the liturgy has been updated in line with changes in the English language and with ecumenical developments as well as to take account of the insights of liturgical scholarship.
All these different Anglican prayer books still have a family likeness, however, as they are all descended from the 1662 book and, beyond that, to its ancestors used by the saints and apostles. As we use the liturgy, we stand together with the "angels and archangels and with all the company of heaven". A good place to be.
PMP