What does this $(\gamma _{0.clothes} + \gamma _{1.clothes} * \ kids)$ term mean in my linear demand equation?
If you multiply out any one of the equations,

$$w_{clothes}= \beta _ {clothes} + \alpha_{clothes} \cdot \ kids + (\gamma _{0.clothes} + \gamma _{1.clothes} \cdot \ kids)\cdot\ income + u_{clothes}$$

$$=\beta _ {clothes} + \alpha_{clothes} \cdot \ kids + \gamma _{0.clothes}\cdot  income + \gamma _{1.clothes} \cdot \ kids\cdot\ income + u_{clothes}$$

it is evident that you have "kids" and "income" as regressors, plus an interaction term