。
The LDP rode on the coattails of Koizumi's popularity.
Literal
LDP [subject-が] Koizumi-popularity [genitive-の] coattails [object-に] rode.
A political commentary using the idiom 尻馬に乗る (しりうまにのる, 'to ride on someone's rear horse' = to ride someone's coattails, to take advantage of another's momentum). 自民党 (じみんとう) is the Liberal Democratic Party, Japan's dominant political party. 小泉人気 refers to the popularity of Koizumi Jun'ichiro, who was PM from 2001-2006 and known for his populist appeal. The idiom carries a slightly negative connotation — opportunistic bandwagoning rather than earned success.