I believe

this is a better link that explains what a structure is in the context of mathematical logic.

If I understand correctly, g is an isomorphism between A and B if

(1) g is a bijection between |A| and |B|,

(2) for every functional symbol f and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle g(f^A(\vec{a}))=f^B(g(\vec{a}))$, and

(3) for every predicate symbol R and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle R^A(\vec{a})$ iff $\displaystyle R^B(g(\vec{a}))$.

The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).