This used to be covered in textbooks.
A fairly recent article about it is "Why do forces add vectorially? A forgotten controversy in the foundations of classical mechanics" by Marc Lange in the American Journal of Physics 79(4) 380-388 (2011); http://dx.doi.org/10.1119/1.3534836
And there are two common answers.
In dynamics you can used Newton's Second Law $$\sum_n\vec F_n=m\vec a$$ and the fact that mass is a scalar.
In statics you can attempt a symmetry argument where you try to argue that the sum of two forces couldn't point in any direction other than the vector sum.
Of course, if you use the former someone can ask why mass has to be a scalar. And if you use the latter they could ask why vector addition is pairwise, or associative, or so on.
Eventually we say that physics is about understanding the universe so you make theories that have models that can be compared to experiments and observations and when the comparison is good that's a success.
And sometimes we will talk about torques and moments of inertia to challenge mass being a scalar (and torques might more rightly be a bivector or a 2d plane instead of a vector or a 1d line). Or we might say that the sum of the force due to A alone and the force due to B alone is different than the force when both A and B are there if the presence of both A and B causes additional forces beyond what each would produce if it were alone.
But even so you still might want to break the vector down into linear combinations of basis vectors, so you still might end up with the exact same math even in these more complicated situations.
This answer to a related question goes into more details.