Let $f$ be a differentiable function and $c$ a number, The set of points $P$ s.t $f(P)=c$ and $\nabla f(P)\ne 0$, is called a surface. (The author didn’t give any informations about the domain and the range of this function.)
Let $X(t)$ be a differentiable curve, we say that $X(t)$ lies on the surface if $f(X(t))=c$
I understand the first line of this definition, but i don’t know why $f(X(t))$ Should equal $c$, if $X(t)$ lies on this surface then $$X(t)=(x_1(t),...,x_n(t))$$ Where $(x_1(t),...,x_n(t))$ is a point on this surface
A differentiable curve is just a set of points of the form X(t), continuously organized by the parameter t, where X is a map that takes a real parameter to a point.
A surface is also a set of points, and a point is said to be in the surface generated by f and c if it satisfies a condition of the form f(P)=c, where f is a function taking points to real numbers, and c is a real-valued constant.
A curve is then said to lie within a surface if all the points in the curve are also points in the surface. Since points in the curve are just points of the form X(t), the condition that the curve lies on the surface defined by f and c is just the condition that f(X(t))=c for all allowed values of the parameter t.
Please let me know if you are still stuck, I’m glad to keep helping you through this!!
UPDATE: Just wanted to make one additional remark: In my original answer I gave some intuition for the right answer, but I never really addressed the problem with the one you gave. For the sake of making this as helpful as possible, I’d like to briefly do that now. You initially wrote X(t)=c, and this isn’t actually a well-defined statement! Remember that, for some t, X(t) is a point. On the other hand, c is a real number, and saying that X(t)=c is saying that a point is equal to a number, which isn’t a meaningful equality.