"If in a few moments this table begins to turn round a little, don't put it down to your inroads into the champagne. I don't wish you to do yourself an injustice."
Chesterton is one of those oft-quoted sources for wise, interesting, and often humorous quotes.
The table is about to turn. To turn the tables can be used as a figure of speech meaning a reversal of fortune, but we would need more information to know whether that is the case here. As I see the table begin to turn, I might be inclined to think that the champagne is catching up with me, that the shift is an illusion of the early stages of intoxication.
The inroads provide the way into something, the beginning. The inroads into the champagne would be the tingly effervescence and light-headedness of being tipsy, or slightly drunk.
He means that I should not write this turning of the table off as mere drunkenness, which would be underestimating my abilities ("an injustice"). Apparently the turning of the table will be real and significant.