My understanding of the argument is that you have to assume you keep playing after you lose, so your stake is dictated by the risk. For the example in the article, the potential upside is 110% profit and the potential downside is 100% loss, so the optimal stake is relatively conservative in order to prevent any loss from impacting your future ability to invest.

Imagine you get the expected result of one win and one loss and had staked 5%: you'll end up with 1.055×0.95=1.002 wealth. If you had picked a 50% stake, you'd be sitting at 1.55×0.5=0.775 wealth instead, since the loss more than erases your winnings.

The unstated contrast is to models where e.g. there are only a limited number of investment opportunities, losses aren't total, or there are capital infusions.