Answer:
The Catholic Church needed priests who could say mass and read the Bible, and monks who could copy Bibles. It kept monastery schools to train these people. Most of the schools in Western Europe of the Early Middle Ages were monastic schools of this type.
I want to point out, however, that the Catholic Church was not the only institution teaching people. The Byzantine Empire had a system of state run primary schools that operated down to the village level and taught both boys and girls, partly so all their soldiers could be literate. Also, the best schools in the West during the first 300 years or so of the Middle Ages were run by the Celtic Church, which was later absorbed by the Catholic Church. For example, around 550 AD, Cor Tewdws (Theodosius College), in Wales had 2000 students. And there were state run schools; Beverley Grammar School, in Yorkshire, was opened in 700 AD as a state run School, and remains open to this day.
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