Let's take a number like 3 divided by 7.
That would be 0.428571428571428571 with 428571 repeating infinitely.
That would be analogous to periodic tiling.
You put shapes in a certain order and you can tile infinity by just repeating those shapes in that order.
It might not be 1 shape, the arrangement might be quite complicated, but it still repeats.

Now think of a number like Pi.
The first few digits of pi are:
3.141592653589793238462643383279502884197... but it continues on forever.
You will never find a series of digits that's finite in length that makes up Pi by just repeating it infinitely.
This is like aperiodic tiling, you will never find a finite group of shapes that makes up the whole thing by repeating it infinitely.
The arrangement of any part is always at least slightly off.

Now math guys have known about aperiodic tiling for a while, but they found it worked with only a few different shapes put together.
They did more math and didn't find anything that said they couldn't do it with just 1 shape.
That's why the hat is exciting to the math people. Knowing something can exist and actually finding the thing are two very different things in math. So they were really excited when they found that it was a relatively simple shape.
It could have been some monstrous shape that takes up 200 GB, or something that would only be found in the year 2723 or something.
They're just glad we have it in the here and now and on a regular screen.