A great way to think about the Central Limit Theorem is through this graphical display, also called the “Quincunx”.
The simulator in the video, shows balls falling from above. At every downward step, each ball makes a random decision about whether to go right or left (in the “quincunx" machine balls hit against some knobs and bounce around, to the left or to the right).
As we run this experiment over the time and as we keep track of where the different balls have landed, at first it seems like they can land all over the place. But after a while, the distribution of where they land will approximately look like a normal distribution, a bell shape.