The place to start with an answer here is that the Bible was not “compiled by a council of sorts that went through a lot of books and picked specific ones to canonize”. That never happened.

The process of canonisation, or canon-formation, is just – a process. One that took (in the case of the New Testament in particular, which is essential if we want to discuss a Christian concept of canon), several centuries, not really being ‘finally fixed’ until the 4th century, and not settled decisively by a council unless you consider Trent in the 16th century to be it.

You see the emergence of a ‘functional canon’ at least in the late first/early second century, in the writings typically grouped as the Apostolic Fathers. These writings tend to make allusions, and in some cases citations, of New Testament texts, and treat the Apostles as authoritative figures. Polycarp appears to allude to several Pauline letters, and the gospels. Papias writes explicitly abut the authorship of gospel texts, and their treatment as authoritative.

What ‘functional canon’ begins to look like, is that Christian communities used certain texts, publicly read certain texts, and treated certain texts as authoritative. And the core of that canon was the four gospels and the Pauline writings. The use of NT texts begins to rival and outpace OT texts in early Christian communities.

By the time we get to Justin Martyr (ca 150s) and Irenaeus (ca 170s), there is a clearer practice of citation going on, and treatment of NT books as “scripture” bearing authority.

Our first canonical “list” is the Muratorian fragment, often dated to ca. 180. It affirms a list of 22/27 NT books, and fails to mention Hebrews, James, and 1 & 2 Peter. It explicitly rejects Shepherd of Hermas, which some early authors treated authoritatively, and likewise the Epistle to the Laodiceans, as a forgery. It also mentions Apocalypse of Peter as a book disputed about.

Once you get to the 4th century, lists become more frequent, and more finalised. Eusebius has one (Eccl Hist 3.25), which lists four categories: Recognised, Disputed, Spurious, and Heretical. Athanasius’ 39th Festal Letter (367 AD), contains a canonical list recognising 27 NT books, and is the first to ‘match’ the contemporary NT canon. From the late 4th century on there is not a great deal of disputation about canonicity of books themselves.

Who claimed the authority to pick these books & how did they do that?

In a sense, nobody. Rather, Christian communities recognised and received certain books as authoritative, and used them for corporate liturgy and private instruction. Those books became the canon. Where there was variation, it was between communities – heterodox and alternative versions of Christianity tended to use and give credence to other books, but the very fact that a relatively consistent and universal church existed across the playing field, led to a relatively consistent idea of which books were in. Which is why by Eusebius’ time, he is talking about some books as ‘disputed’ – books which not everybody everywhere is ready to agree to, because of some hesitations – often about apostolic authorship, sometimes about content.

if any of the Church's early fathers ever wrote anything about how & why they chose the Bible

Again, not really, because it didn’t happen the way you think it did. When they do comment on things like this, they mainly record opinions such as, “this is not to be read”, or “this is not accepted everywhere”. When they judge a book not to be acceptable, they sometimes provide grounds. For example, as mentioned above, Shepherd of Hermas comes to be rejected, because it is seen as a late, post-apostolic composition. Known or believed forgeries, such as the Epistle to the Laodiceans, also rejected for that reason.