The answer to your question is in the six lines that immediately follow the sentence that you quoted. You start with an indexed family $\{M_i:i\in I\}$ of structures of the same type (linear orders, rings, fields, etc. $-$ technically, they have the same signature). You form the Cartesian product $M=\prod_{i\in I}M_i\;;$ the structures $M_i$ are the factors of this product. You take an ultrafilter $\mathscr{U}$ on $I$, and you use it to define an equivalence relation $\sim$ on $M$:
$\langle m_i:i\in I\rangle\sim\langle m'_i:i\in I\rangle\text{ iff }\{i\in I:m_i=m'_i\}\in\mathscr{U}\;.$ The ultraproduct of the $M_i$ is the quotient $M/\sim$, often written $M/\mathscr{U}$, whose elements are the $\sim$-equivalence clases. Although $M/\sim$ is no longer strictly speaking a product, we still speak of the $M_i$ $-$ the genuine factors of the Cartesian product $M$ $-$ as its factors.