In story text, direct quotes that are laden with brackets are distracting. The insertions defeat the purpose of the quote — to allow the reader to see the exact words of sources. A paraphrase or partial quote is an option when the direct quote simply doesn't work without the bracketed information. A good setup sentence for the direct quote can help too.

In a pull quote like this one, the brackets are even more distracting. The idea of the pull quote is to highlight an especially illuminating statement. Pull quotes should be pithy and clear. Splicing in explanatory text between brackets ruins that effect.

UPDATE: See a related discussion here and a previous post here.