measurments

This comes from math. If you take any number to the power of negative one, the result is 1 devided by that number.

Another way to say it is x^(-1) = 1/x

so mg kg-1 is another way of saying mg * 1/kg or
mg/kg.

Unfortunately mg/kg doesn't make much sense (it would
mean milligram per kiligram). Can you explain the context
of the problem this came from?

Chemical per weight
In this context, I think this means the amount of hydrogen cyanide you can extract from 1 kg of flaxseed. So if you get 187 mg kg-1, and you have 10 kg of flaxseed, you should get (187 mg/kg) * (10 kg) or 1870 mg of hydrogen cyanide.