Abbreviare to abréger is a very regular sound change, it went something along these lines.
[a.bbre.vi.a:.re] > [a.bre.vja.re] > [a.brɛ.vɟær] > [a.brɛ.vɟʒier] > [a.bre.dʒier] > [a.bre.dʒie] > [a.bre.ʒe]
Basically the key is desyllabification of [i] to [j] and then its fortition and assibilation to [dʒ]. The [v] preceding a consonant is lost regularly. In the Middle Ages the change was in its middle (abrégier with <g> being pronounced [dʒ] as in Italian and contemporary Spanish). Around this time, the word penetrated to English, which did not have the distinction between closed and open [e]/[ɛ] and chose to interpret the former as a short [i]. The [dʒ] sound was then represented by <dg>, this might have been later and under the influence of the similar word bridge.