This can be explained as follows. Mathematically, given two variables X and Y, their Correlation is defined as the
covariance(X,Y)/(Standard Deviation(X)*Standard Deviation(Y)).
In other words, the correlation is proportional to the the covariance of the two variables. The divisor in the equation acts has a scaling effect on the covariance so that the resulting correlation will lie between -1 and +1.
So, all other things being equal, reducing the covariance will reduce the correlation. The effect of having similar school achievement is to reduce the covariance between IQ and school achievement. For example, given a wide range of IQ's, if school achievement is similar then school achievement doesn't co-vary with IQ, i.e. there is a relatively random relationship between achievement and IQ, i.e. the correlation is close to zero indicating (relatively speaking) no relationship.
On the other hand, given a wide range of IQ's, if school achievement is also spread over a wide range then correlation can still take any value between -1 (a negative relationship and +1 (a positive relationahip) including 0 (indicating no relationship)
Getting back to your question, it is the reduction in covariance that is important here rather than the reduction in variance.