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"That" refers to "the introduction"

The authour could have said "this" and it would have made just as much sense.

"That" usually refers to something in the past.
"This" usually refers to something in the immediate present.

In my opinion the word you have highligted is not necessary.
In the example below I have left out "that and", the sentence is much easier to read.

With that much introduction, a ready acquiescence to terms and a couple of sovereigns flung upon the table, he top up his quoters in the inn.

"he top up his quoters in the inn." This does not make sense There is no such word as quoters perhaps this is a typographical error.

If i hold a pen in my hand it becomes "this" pen in my hand
if the pen is on the table it becomes that pen on the table.

English is a very difficult language because it is made from so many other root languages, even for native speakers like me

I believe that "top up his quoters" is meant to be "took up his quarters". The first portion probably refers to an introduction from a previous sentence. Without that, it deoesn't make sense to me.