Why do HFs “risk off” when trades move against them?

There are a few reasons:

1. If you're leveraged, when you lose money your leverage ratios (or VAR or choose your risk metrics) get worse. Let's say you're 4-to-1 leveraged, and you lose 10% of your gross book. Now you're at 8-to-1 leverage. If you want to keep your leverage constant, you have to reduce risk to adjust for the fact that your equity has been reduced. 

2. If PMs at a platform lose money, the optionality of their payoffs increases, incenting them to take more risk. For a (very) simplified example, let's assume that if you, the PM, lose money over a Jan-Dec calendar year at a multi-manager, you get fired, but if you make money, you keep a % of the PnL. If you're highly profitable year-to-date, you're going to be cautious about losing money as it comes out of your bonus; your risk/return incentives are going to be balanced, and mostly aligned with those of the platform. But let's assume instead that you're losing a lot of money year-to-date. It doesn't matter how much you lose by year-end; the outcome is the same, you get fired. Thus, you have a massive incentive to take on a ton of risk (for instance, betting heavily on a theme) to attempt to get back to positive returns. You're no longer aligned with the platform, as the risk of further downside is meaningless to you. Thus, the platform needs to keep an extra close eye on you and reduces your risk allocation to offset your incentive to take on more risk.