Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.

Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has:
and its generalization the indefinite orthogonal group has:
where S is symmetric. So for the symplectic group we have matrices Y such as:
where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.

$O_{T}IO=I$

$O_{T}SO=I$

$Y_{T}AY=I$

Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.