What does Hayek mean at the end of the first paragraph of The Use of Knowledge?
Your marginal rate of substitution between (say) eggs and wine is the number of eggs you'd be willing to trade for one additional bottle of wine.  This marginal rate of substitution can reflect your preferences (if you're a consumer) or the state of technology (if you're a producer who can, for example, substitute three eggs for one wine in your production process).

Suppose your MRS is three eggs per wine and my MRS is two eggs per wine.  Then there's an opportunity for a mutually beneficial trade:  You give me two and a half eggs, and I'll give you a wine.  You thought the wine was worth three eggs, and you got it for two and a half --- that's a win.  I thought the wine was worth two eggs, and I sold it for two and a half --- that's also a win.

Whenever your MRS differs from my MRS, there's an opportunity for a mutually beneficial trade.  

Therefore, if our MRS's differ, it means that an opportunity for mutually beneficial trade has been missed.  

So in order for resources to be used efficiently --- that is, in such a way that all opportunities for mutually beneficial trade have been exploited --- the MRS between any two commodities must be the same for you as it is for me or anyone else, regardless of how we're using those resources.