I was reading the book: Mathematical Methods in the Physical Sciences by M. Boas and I came across this statement; 
I wasn't quite sure why this was the case. Is it because in the curvilinear coordinates angles between the two coordinate axes are dependent on the position, hence one cannot use the projections to get the transformation matrix form (x,y,z) to (x',y',z') - like in Cartesian systems related only by rotations?
It might be though that I don't understand the term linear transformation properly?
A linear transformation is one that obeys the rule $f(\alpha x+\beta y)=\alpha f(x)+ \beta f(y)$. And we have clearly not $\sin(\alpha x+\beta y)=\alpha \sin(x)+ \beta \sin(y)$. But differential operators (like partial derivatives or differentials) are. The difference is that here the variables don't live on the reals or in a vector space, but that they are functions. So we have : $\mathrm{d}(\sin(\theta)+\cos(\theta))=\mathrm{d}\sin(\theta)+\mathrm{d}\cos(\theta)$. Even for functions that are vector-valued a change of variables can give rise to a linear operator (in your case the Jacobian of the change of variables).