So, imagine you have a big square (let's call it a matrix). You can put numbers in each little square inside the big square.
Now, if you draw an imaginary line down the middle of the big square (like a mirror), and the numbers on the left side of the line are the same as the numbers on the right side of the line, then it's a symmetric matrix.
Basically, it's like folding the matrix in half and everything looks the same on both sides. Kind of like when you make a paper snowflake and everything is the same on both sides when you unfold it.