That obscure ‘would’… again

Speak up, giugno, pagina 31: “You know, when I go back to when I was studying, I would go to the public library, I would be taking out books that had already gone through that editorial process.” Perché viene usato would+be+ing form? È una forma particolare del futuro continuo? Grazie anticipatamente. Alberto

The use of “would + the infinitive” is a way of expressing a habit in the past, e.g. “When we were children our grandmother would take us to the zoo”. It is the same as “When we were children our grandmother used to take us to the zoo”.
When Caroline Kennard says “I would be taking out books” the meaning is very similar to “I would take out books” or “I used to take out books”. She uses the continuous form to express the idea of it being a continuous, ongoing action. She does this for narrative, rather than grammatical, reasons. The meaning is the same.