In the short term, the reason that the late Queen's funeral is being held at Westminster Abbey is that the plan says it will be.
Sam Knight's excellent article in the The Guardian, which has been right on every detail to date, assumes without further explanation that the funeral would be held at the Abbey. Planning for this ceremony has been undertaken for over half a century and the location must surely have been one of the first decisions taken, because so much depends upon it. Rewriting almost the entire plan at short notice, with many of the key figures being grief-stricken and busy at other events, would be an extremely risky decision.
Of course, that raises the long-term question of why Westminster Abbey is the place in the plan. That is answered by CDJB's post, and Stuart F's important comment to it about policing.
I would add a couple of minor points to their overview. Firstly, the coffin will lie in state at Westminster Hall, so moving it the 200 metres or so to the Abbey is surely less complicated than moving it through the West End to St Paul's, on a day when security resources are already stretched to the limit. Secondly, there is an ecclesiastical difference between St Paul's and Westminster Abbey. St Paul's is the cathedral of London and primarily exists to serve the spiritual needs of that city. Westminster Abbey is a Royal Peculiar - it stands outside the parish system and is part of no diocese. So it has come to specialize in hosting events of national significance. If you're going to close a church building for ten days without notice, it's probably less disruptive to close the Abbey than a diocesan cathedral, as many of the senior cathedral staff are double-hatted and also have diocesan responsibilities. I doubt that either of these points were conclusive though.