For GAMs in regression problems, in order to depict a non-linear relationship between the predictors and their response, we start off with the multiple linear regression model $y_i = \beta _0 + \beta _1x_{i1} + \beta _2x_{i2}+...+ \beta _px_{ip}+ \epsilon _i$.We then replace the linear component ($\beta _jx_{ij}$) with a non-linear function $f_j(x_{ij})$ so that the original multiple linear regression model becomes $y_i = \beta _0 + f_1(x_{i1}) + f_2(x_{i2})+...+f_p(x_{ip})+ \epsilon _i$. A separate non-linear function $f_j$ is applied to every predictor variable in the regression model, then added together. This is why it is considered an additive model.