。
She gained a high standing in that town.
Literal
She [topic-は] that town [in-での] standing [object-を] obtained.
信望 ('confidence and respect, public trust') is a slightly bookish word for the kind of standing that comes from being respected by others — earned reputation, not just visibility. での is the compound particle で + の: で marks the location of the standing, then の attaches it as a modifier of 信望. This composite-particle use lets you turn a で-clause into an adnominal modifier ('the standing in that town'). 得る ('obtain, gain') is more formal than もらう or 手に入れる; common in writing for abstract acquisitions like reputation, knowledge, opportunity.