In what sense does the universe have an outer edge?

There is a surface more or less at 13.7 billion light years from us which is where we see back to the big-bang (looking back in time, if you define "now" in a global way, although there is no reason that we should't define "now" by what we are seeing "now", i.e. along a past light cone). This surface is analogous to a black hole horizon, except it surrounds us instead of being localized in a region.

This thing is called the "cosmological horizon", and the general idea of the holographic principle suggests that everything inside the cosmological horizon is described by oscillations of this horizon. This is hard to make precise because the horizon has a finite area and growing, and so has a finite maximum entropy associated with it (which is growing), and this is paradoxical seeming, because it suggests that the Hilbert space for our universe is growing.

The number of states in a quantum mechanical system can't increase, so this leads many people to renounce the idea of string theory in our kind of universe, choosing instead to describe the dynamics in terms of the asymptotic future, where presumably the universe will vacuum decay to a supersymmetric state. This is one approach, another is to try and formulate a real theory with a finite Hilbert space. I think a possible third approach is to consider finite area horizons as somehow density-matrix like, so that they, unlike black holes, have fundamental decoherence. Nobody knows the answer, and this is the major unsolved problem of string theory today.