How do multiple bonds introduce rigidity into molecules?

Multiple bonds ....

When you hear "multiple bonds" you think "double" and "triple" bonds, as opposed to single bonds. Around single bonds, there is free rotation. No so with double or triple bonds because of the pi-bonding.

depends on what you mean by multiple bonds. Mike A interprets that to mean sharing of more than one electron between two specific atoms. I would interpret it to mean several distinct bonds between different specific atoms. Rigidity arises from conflicts between the different constraint to motion inherent to any individual shared attachment. Each bond would retain a rotational movement potential but have limited lateral or in-out motion. Well, when two atoms are attached to a third atom, the freedom to movement can get constrained by the rotation of one smacking head on against a resistance to move laterally by another.

You only truly get rigidity when the network is in a three-dimensional structure. lines and planes retain that sliding or rotating aspect that a third orthogonal structural strut makes impossible. Difference between a pencil, a sheet of paper, and a box.

The double bond consists of:

1.A sigma bond formed by end on overlap of atomic orbitals. Sigma bonds on their own allow for rotation about the bond.

2.A pi bond formed by sideways overlap of atomic p orbitals. Rotation about the bond in this case would cause it to rupture so the sigma/pi combination is rigid.

Explained nicely in this video:

https://www.youtube.com/watch?v=pT8nrBrTOm4