How does the fact that banks hold a fraction of their assets in a central bank put pressure on companies to grow exponentially? I'm curious, I don't understand the connection.

Yes.

But unfortunately? Without safety measures like the Fed enabling such, economies have a history of imploding every fifteen years or so on the basis of insufficient currency circulation. Oh wait, that's where we are today... Carry on!

Commodity-backed currencies also have a problem - they implement their own fractional reserves as economies do grow. When they shrink? End-users learn to trade in the notes for the commodities, shrinking the pool of currency in circulation.

Fractional reserve banking is one part of it, but the really horrible things occurred when we left the gold standard so almost unlimited debt can be created. People started investing in stocks instead of bonds, which is probably part of why Glass-Steagall was repealed.

Could you expand on this for someone with very limited economic intuition? (me)

With higher fractions of reserve the bank is also a more stable institution. So it may be just a symptom, not the source.