Extract the symmetric matrix built-in another matrix
Maybe a bit simpler:
M Transpose[M]
Comparing with Henrik's answer:
M Subtract[1, Unitize[Subtract[Transpose[M], M]]] == M Transpose[M]
True
Maybe this is what you are looking for. I interpreted your question as if you want to replace all nonsymmetric entries of the input matrix by zeroes and as if the input matrix is not necessarily binary.
A = M Subtract[1, Unitize[Subtract[Transpose[M], M]]];
A = Floor @ Symmetrize @ M;
TeXForm @ MatrixForm @ A
$\left( \begin{array}{cccccc} 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} \right)$
Note: This is much slower than the methods in the answers by bill s and Henrik Schumacher.