Mechanism by which $lacI^{d}$ is a dominant mutation, impairing the function of normal copies of the Lac Repressor
The lac repressor act as a tetramere molecule and requires all 4 of the subunit to be able to bind DNA to act on the operon and repress β-galactosidase expression.

The "all 4" is the key here, if any of the 4 subunits is unable to bind DNA then the whole complex cannot attach to the operon. The lacId mutation produces a repressor subunit that cannot attach DNA, yet can tetramerize with other lac repressor subunits, therefore impeding WT subunits to bind the operon.

For your question on the protein structure. Mutations can occur at different locations in a gene and therefore impact different proteins domains. Proteins contain different type of domains, such as an active site for an enzyme or a binding sites for attaching to the DNA. A mutation might act on one of the domain while leaving others fully functional.

In your case this is exactly what happens, the mutation is in the protein region involved in binding DNA but does not affect the region involved in the protein-protein binding with other subunits.