how to explain the term “Compensated law of demand” in a plain way?

The compensated law of demand is a tool we use to analyze the decomposition of substitution and income effects. We take a price change which changes the relative price ratio (the slope of our budget constraint in a simple two good example) which leads to a new tangency point on a new 'rotated' budget line. For simplicity, if we consider the tangency condition: $$ |MRS| = frac{MU_1}{MU_2} = frac{p_1}{p_2} $$ Then as $p_1$ rises, for example, the value of our MRS defined as the slope of our indifference curve would rise at the optimal point. So our new tangency would be higher up on this new lower indifference curve. What we would then do is ask the hypothetical: Under the new prices, how much income would leave the consumer just as well as off before the price changes? This is the idea of compensated demand. What would we do in practice is essentially shift the budget line up (or down) until it is tangent to the original indifference curve. This gives us the pure substitution effect abstracting away from the change in purchasing power.

Since you are reading from an advanced microeconomics textbook I imagine you are familiar with Marshallian and Hicksian demands, these concepts are very closely related to how we compute this hypothetical budget constraint. Sadly that is the simplest way I can think of explaining it as it is an 'advanced' concept. In practice, we don't usually use the tangency condition to arrive at these results but it helps in simplifying the concept.