OK, so what we discovered was that if we look far into the universe, all stuff is moving away from us (we can measure that using the famous red shift). Additionally, the farther something is, the faster it is moving away from us.

Now the simplest explanation would be that the Earth just happens to be exactly in the middle of this expansion. However, it is much more reasonable to assume our location in the universe is not special in any way and that you'd see things moving away from you if you did these measurements anywhere else. If that is so, the universe is expanding, everywhere. There's more space in the universe every second and there's still the same amout of matter, hence it is becoming larger, emptier and colder.

The next step is to look back and think of how the universe looked in the past. Since it's getting bigger and bigger, it must have been smaller before and if you go back in time enough you'll find a tiny universe that still has the same total amount of matter in it, just densely packed into possibly just a single point. Hard to say what preceded that moment, but we can predict a universe which started as an incredibly energetic singularity which exploded out and has been growing ever since. We call that moment the Big Bang and if anything, that is the centre from which everything is expanding. Not found somewhere in space but in time.

Back to the balloon analogy, the centre would be the deflated little thing you start with. Maybe a better analogy would be the little clump of molten glass at the end of a glassblower's pipe. He begins to blow, the big bang happens, expansion starts. Fast forward 13.7 billion years (which happens to be today) - the glass has expanded into a large spherical object and there's some tiny people living on it.

They only live on the surface of the glass and the sphere is so huge (or they are so tiny) that they can't even tell that their world is spherical. They measure distances to some other specs littered across the glass and find that they are all moving away from them, faster the farther they are. Their universe is expanding, but where is the centre of expansion? They cannot point to it, because they only live in two dimensions, fully defined by the sufrace of the sphere. But if they could point in a direction perpendicular to all the spacial dimensions they know, they could point to the point where the sphere started, long in their past.

So the right question is not form "where" the universe expands but "whence". The Big Bang. The very start. Somewhere far in time, which is just another direction, perpendicular to up, left, forward.

With the balloon analogy it's not about the center of the volume, we ignore the volume and assume that the surface is a 2d universe. That's what's impossible to find the center of. I don't really like that analogy though personally so I'm not going to discuss that one further.

Just think about it this way: the observable universe can only be so big (because when the expansion between two distant enough objects is faster than the speed of light/causality, they no longer have a means of interacting). We don't observe any sort of obvious boundary to the universe within our visible portion that we might be able to assume a center based on. So it's not that we know that there isn't a center (afaik, someone correct me if I'm wrong), it's that it's likely impossible know that there is, let alone find it from our position in the universe. So, we might as well assume that it's all relative.

Imagine you woke up on a raft in the middle of the ocean on an alien planet. It's foggy, you can't see stars, you can't see any landmarks at all. There are other things floating in the water too though. There might be a geometric center to that ocean, but you can't see it, and you have no other hint at where it is. For all you know, the entire planet is ocean and there's is no center to find. This is sort of the situation we Earthling are in now, except that at least the the rafter can drift and perhaps eventually find and map out a coast. Because our space-time is expanding, our observable universe will never be bigger than it is now.