My maternal grandmother insists on wearing that old-fashioned dress.

Literal

Maternal-line [possessive-の] grandmother [topic-は] that old-style dress [object-を] wear thing [to-に] insists

母系の祖母 = maternal grandmother (母系 = maternal line). 旧式 = old-fashioned/outdated (旧 old + 式 style). 固執する (こしつ) = to insist on/cling to — a strong word implying stubbornness. ことに固執する = 'insist on (doing X).'