It seems to me that you can reason like this. Consider the boundary B of a small neighborhood N of $F$. Then this boundary B is connected if and only if F is 1-sided. Now, conisder the intersection of J with B. There are even number of intersections, since B is the boundary. So you can throw the part of J that does not belong to the neighbohood N an close it to a connected curve by segments in B (we assumed that B is connected).

We want to prove that in the case F is not one-sided, we may replace J by a curve J' that is contained in a small neighborhood of F and interesects F in the same way as J. By assumtion F is one sided. Consider the boundary B of a small neighborhood N of $F$. Since F is one-sided, B is connected. Now, conisder the intersection of J with B. There are even number of intersections, since B is the boundary. So you can throw the part of J that does not belong to the neighbohood N an close it to a connected curve J' by segments in B (we assumed that B is connected). This explanes the words written in bold.