We consider the problem of short- and medium-term electricity demand forecasting by using past demand and daily weather forecast information. Conventionally, many researchers have directly applied regression analysis. However, interpreting the effect of weather on the demand is difficult with the existing methods. In this study, we build a statistical model that resolves this interpretation issue. A varying coefficient model with basis expansion is used to capture the nonlinear structure of the weather effect. This approach results in an interpretable model when the regression coefficients are nonnegative. To estimate the nonnegative regression coefficients, we employ nonnegative least squares. Three real data analyses show the practicality of our proposed statistical modeling. Two of them demonstrate good forecast accuracy and interpretability of our proposed method. In the third example, we investigate the effect of COVID-19 on electricity demand. The interpretation would help make strategies for energy-saving interventions and demand response.