Why would composers of polyphonic music, such as Leonin, use a pre-exisiting chant as the basis for their work?

In the early days of the medieval Church, music was set to have only one part of singing. Eventually, singing the same as everyone else all the time became boring, and the church decided that maybe two parts of singing were ok, as long as they remained within reason.  

In order to achieve new music while adhering to the strict musical rules of the Church, the composers, like Leonin, added additional vocal parts to what was previously a single line of church singing.

Explanation:

Polyphony ocurrs when two or more independent melodies are sung or played simultaneously.

Everything new had to be based on something old. This meant that any new composition had to be based on a pre-existing composition, such as church chants. This school of thought is known as Ars Antiqua.

Many times, the original song was sung at an extremely slow pace, while a new and faster tune was added with more tones in a higher pitch. This new type of multi-part song was called organum.

This is because polyphonic music can only be composed by using two-voiced settings of soloistics chants.

This was first credited to Leonin(active between 1165 - 1185), who used two-voiced settings of soloistic chants to composed music during church mass.

Explanation:

The preexisting chants, also known as "a cantus firmus" OR "fixed song" is the basis of a polyphonic composition.

Early polyphonic compositions almost always involved a cantus firmus, typically a Gregorian chant, although by convention the term is not applied to music written before the 14th century.

The earliest surviving polyphonic compositions, in the Musica enchiriadis (around 900 AD), contain the chant in the top voice, and the newly composed part underneath; However this usage changed around 1100, after which the cantus firmus typically appeared in the lowest-sounding voice.

Later, the cantus firmus appeared in the tenor voice (from the Latin verb 'tenere', to hold), singing notes of longer duration, around which more florid lines, instrumental and/or vocal, were composed or improvised