Questions & Answers about Она попросила об одолжении.
О and об are two forms of the same preposition, meaning about/regarding in this kind of construction.
Russian uses об before words that begin with a vowel, mainly for ease of pronunciation:
- о брате
- об отце
- об одолжении
So об одолжении is used because одолжении starts with о.
You may also see обо in a few fixed or traditional combinations, such as обо мне.
Because after о / об, Russian normally uses the prepositional case.
The dictionary form is одолжение, but in the prepositional singular it becomes одолжении:
- одолжение → nominative
- об одолжении → prepositional
This happens because nouns ending in -ие usually change to -ии in the prepositional singular:
- здание → в здании
- сомнение → в сомнении
- одолжение → об одолжении
So this is a normal case change, not an irregular phrase you just have to memorize.