Looking for easy applications of fractional cascading
Chazelle and Guibas wrote a paper describing some of the good applications that can be made with Fractional Cascading: Link

Some of the most interesting applications are:


Intersection of a polygonal path $P$ and a line. Given the path $P$, pre-process it and allow queries of the form: Given a line $l$, report all the intersections of $P$ and $l$.
Orthogonal Range Search. Given points in $R^d$, and a $d$-orthotope. Return the points that lie in the orthotope.
Planar point Location. Locate a point in a planar subdivision.


You do have to give a bit more background as this problems tend to be geometrical in nature. Fractional Cascading is really only good when tackling higher dimensional problems.