Given $n$ independent nonlinear equations in the $n+1$ variables $x_1,\dots,x_{n+1}$, is the solution always of dimension 1?
For example, if $n=1$, $x_1\sin(x_1+x_2)=0$ gives a curve in $\mathbb R^2$.
It is not clear to me whether this is something general are if it is only true under some conditions.
Yes, this is something general; but no, this does not always happen. Counterexample: $x_1^2+x_2^2 = 0$, which is only true for $(x_1,x_2) = (0,0)$. This is because $(0,0)$ is a so-called critical point of the function $f(x,y) = x^2+y^2$. For more information, see Wikipedia: Level set. Having more than one (nonlinear) equation would then be equivalent to finding the intersection of different level sets.