Sol's answer is a good one, but here's a way to illustrate it for yourself.
Get a balloon and a marker. Place some dots on the outer surface of the balloon. The balloon's outer surface represents the spacetime fabric of our universe, and the dots represent galaxies and galaxy clusters. You can even mark one dot with MW for Milky Way (our galaxy).
Now, slowly inflate the balloon and observe what happens. From the point of view of MW, all the other galaxies and clusters appear to move away -- likewise, from the point of view of another galaxy, the MW and all others would appear to move away. So, the idea is that no matter where you are in the universe, you will observe everything to be stretching away from you and could easily conclude that you're at the "center".
This is an illustration of what's known as the
Cosmological Principle.
Two important points to note about this example:
1. People will often ask, "What's inside the balloon? Isn't that where the 'center' is?"
The answers are "nothing" and "no." Remember that in this model the entire universe is the stretched fabric of the outer surface of the balloon. The question of what's "inside" isn't relevant, unless you wish to speculate about hypothetical extra dimensions of space, etc.
2. Note that the dots on your balloon do not actually move along the surface of the balloon; rather they move
with the balloon's surface as it stretches. In addition, dots further from the MW dot will appear to stretch away faster than nearby ones.
This points out a common misconception about big bang cosmology: that galaxies and galaxy clusters are actually moving
through space. They aren't (for the purposes of an expanding universe, that is), instead they are being carried along with the spacetime fabric of the universe as it is undergoing an expansion or stretching.
Lastly, about the further points moving faster as compared to the closer ones: this is an illustration of
Hubble's Law, which was one of the first ways we quantified the age of the universe to be many billions of years (roughly 13.7 billion years by current estimates).
I hope this helps. Have fun with your balloons!
