I would think that what it is saying is clear- except whether p is a constant or a function of y. I would assume a constant. To show that z= uv is a solution to that, just go ahead and differentiate z= uv so that z'= u'v+ uv' by the product rule. Then z''= u''v+ 2u'v'+ uv''. (z"+ pz)'= (u''v+ 2u'v'+ uv'')'+ p(u'v+ uv')= u'''v+ u''v'+ 2u''v'+ 2u'v''+ u'v''+ uv'''+ pu'v+ puv'. Now notice that

u''v'+ puv'= (u''+ pu)v' and, since u satisfies y''+ py= 0, u''+ pu= 0. Similarly, u'v"+ pu'v= (v"+ pv)u'= 0.