Basic strength is determined by the ability to donate pair of electrons more readily. (Lewis theory).
In triethylamine, there are three ethyl groups attached to nitrogen. According to +I effect, they are electron pushing groups and increase negative charge on nitrogen. So nitrogen in triethylamine is more ready to give up pair of electrons than in ethylamine where there is only one ethyl group and less +I effect.
So triethylamine is more basic.
