If the matter within the universe is expanding, it has to be expanding into something. What is that something?
The matter in the universe is actually compacting because of gravitation. Most of the reason that matter has not compacted into black holes is that in the early universe the matter was much hotter, and thus components of matter had a lot of kinetic energy, which works against compaction. Ordinary matter readily collides with other ordinary matter (or photons) and the collision radiates away photons, so the matter loses kinetic energy in the process, and so tends to compact into bright dense blobs like stars. (Dark matter collides very rarely, and does not collide with or emit photons at all, so it still has lots of kinetic energy and thus spins in high orbits around massive structures like galaxies; in order to fall into the middle of the galaxy, that kinetic energy has to be lost, and whatever processes dark matter uses to get rid of kinetic energy are verrrry slow).
That is, most of the matter in the universe is in large gravitationally-bound structures. These structures are all moving away from one another, any observer looking at structures from his or her or its vantage point will see the distant structures receding faster than closer structures. The most obvious interpretation of this would be that empty space is being created between the big structures, but since the structures are not themselves expanding, empty space is not being created within the big structures.
In between all these large gravitationally-bound structures the gravitational potentials (which describe the direction things fall and how they appear to accelerate while falling, in the eyes of various observers) are very weak compared to the gravitational potentials near, or within, galaxies. Because it can be seen by certain observers to impart accelerations on objects, the gravitational field has an energy. Matter can receive energy from the gravitational field, and it can also donate energy back to it. (This is a generalized conservation rule in General Relativity), and the energy of the gravitational field is non-uniform. Whatever energy is causing empty space to appear works against gravitation. For example, if the empty space was not being created, the large structures would be closer together and so they would feel mutually steeper gravitational potential gradients -- that is, gravity would bring them ever closer together, merging them, and causing them to compact. That is, gravity would (from our viewpoint) accelerate big galactic clusters (including the one our galaxy is in) towards each other. However, we observe that clusters are accelerating away from one another instead, and that the acceleration is highly uniform with distance.
The simplest way to explain this is to posit an energy field with a small energy value at every point in space; the energy works against gravity by "unfolding" new space from something like a compact manifold. However, the energy value is small enough to be dwarfed by gravitational energy in stars, star systems, star clusters, galaxies, galactic clusters, superclusters, and possibly galactic filaments. So where there is lots of gravitational energy, like in these structures, you wouldn't notice new space appearing. In deep space far from massive structures, gravitational energy is so weak that this unknown (and "dark" as in "dark ages", which are poorly understood bits of history) energy does not suppress the "unfolding" of new space. So, lots of new space appears between galactic clusters. And that new space still has dark energy, so new space unfolds within the new (and empty) space. And so on. The result: exponential growth of the amount of empty space in the universe, all appearing far from big visible structures.
This is called the metric expansion of space because "unfolding" new space has a geometrical equivalence in increasing the number of coordinates, which in one dimension is equivalent to observing a ruler stretching and adding new marks (so a ten million light year ruler in deep deep intercluster space with ten marks on it becomes a twenty million light year ruler with twenty marks on it).
Also, in General Relativity there is a Lorentz contraction of rulers at lower gravitational potentials and a Lorentz expansion of rulers at higher ones, compared to an observer. If we put a ruler near a really massive and dense object it is at a lower gravitational potential, so the ruler would appear smaller to us than its exact duplicate sitting in our observatory; if we put a ruler outside our galaxy, it would appear longer than its exact duplicate here in our solar system. The new empty space that is "unfolding" is at an ever higher gravitational potential (if we put a particle in a volume of expanding space it would have further to fall than a particle we put much closer to the mass-energy the first particle wants to fall towards).
I hope this helps a little.
The balloon may be expanding, but it is expanding into the box/room/whatever. Your explanation simply says the balloon is expanding into itself.
What if you had a huge sheet of paper or tinfoil with enormous surface area and carefully folded it up so that it looked like an inflated balloon, and then added more air to the (pseudo)balloon? Rather than stretching like latex, the material would unfold. If your folds are really really tiny then you might not notice them or think about them until you want to explain why adding air to the (pseudo)balloon didn't rip the unstretchy paper or tinfoil, but instead caused it to appear to expand like an ordinary latex balloon. (Think of an accordion, maybe).
You can't say the matter in the universe was in a ball (metaphorically speaking) and then at some point it began to expand because it has to expand into something, not into itself. Further, what was that point of matter sitting in before it expanded? Was it sitting in emptiness? If so, what was that emptiness contained in?
We have good evidence that earlier in the universe's history the matter in it was in a very uniformly hot dense phase, and it has since cooled down and spread out at the largest scales. At smaller scales -- galaxies, stars, star systems, people, trees -- the matter has compacted, and there are wide variations in local temperature.
The (metaphorical) ball expanded because the "dark energy" and the kinetic energy of the hot matter overcame the forces that try to stick matter together, like gravitation, electromagnetism, and the weak and strong nuclear forces. Little overpressures and underpressures in the early universe led to structure formation and separation respectively. A brief period where the expansion force was very strong compared to the other forces can explain many things we see in the sky with respect to structure dynamics - formal descriptions of this are in the family of Cosmic Inflation conjectures, some of which is solid enough (in terms of consistency and in terms of correspondence with observation and experiment) to call "(incomplete) theory".
What was the early universe "in"? We don't know. Lots of physical cosmology theoreticians have speculated about that, but nobody has much of an idea about how to directly "see" anything older than the process that produced the cosmic microwave background radiation, and have focused on what earlier events would produce patternwise in the CMBR. Maybe some sets of possible "outside the universe" conjectures would produce definitely observable marks on the CMBR, or maybe we will find some other observational techniques. However, at the moment, we just don't know.