The Lagrange multiplier (LM) test statistic is the product of the R2 value and sample size: This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number of estimated parameters (in the auxiliary regression).
First, the squared residuals from the original model serve as a proxy for the variance of the error term at each observation.
(The error term is assumed to have a mean of zero, and the variance of a zero-mean random variable is just the expectation of its square.)
The independent variables in the auxiliary regression account for the possibility that the error variance depends on the values of the original regressors in some way (linear or quadratic).
Conversely, a “large" R2 (scaled by the sample size so that it follows the chi-squared distribution) counts against the hypothesis of homoskedasticity.