Why do exchanges provide ask/bid transaction data like the NYSE and NASDAQ? Bitcointicker.co provides a very slick, very active display that purports to be all the ask/bid transactions on several exchanges (Gemini, Kraken, Coinbase, and others).
Where does it get its data?  AFAIU all the bitcoin transactions inside an exchange are just bookkeeping they do and the only real blockchain-recorded transactions are when you send bitcoin into the exchange, take bitcoin out of the exchange, and all the balancing activity the exchange needs to do to buy and sell actual bitcoin to keep themselves balanced.  So none of these can be actual bitcoin-network transactions (appearing in the mempool), right?
Only thing I can think of is that somehow when you become an SEC regulated exchange you've got to provide public visibility to the market - just like the NYSE or Nasdaq.  Is that the case?  Or is it something else?
I do see from the tag ticker that different exchanges provide different APIs for this.  So my actual question is: Why?  Are they required to?  Because if it isn't a (legal, regulatory) requirement on them I can't see why they do it.
Where does it get its data?

It gets it from the APIs of those exchanges. For example, here's the Coinbase/GDAX API for order book information: https://docs.pro.coinbase.com/#get-product-order-book

AFAIU all the bitcoin transactions inside an exchange are just bookkeeping they do and the only real blockchain-recorded transactions are when you send bitcoin into the exchange, take bitcoin out of the exchange, and all the balancing activity the exchange needs to do to buy and sell actual bitcoin to keep themselves balanced. So none of these can be actual bitcoin-network transactions (appearing in the mempool), right?

Correct, none of this data is from the actual blockchain. It's entirely from the exchange APIs.

Only thing I can think of is that somehow when you become an SEC regulated exchange you've got to provide public visibility to the market - just like the NYSE or Nasdaq. Is that the case? Or is it something else?

I don't think cryptocurrency exchanges are regulated by the SEC since they aren't securities exchanges. It's just a standard feature that traders will expect on an exchange, much like you'd expect a speedometer on your car as a given. Order book information is necessary for traders to make informed trades, and if the exchange didn't make it available it would be missing key functionality.