I tried searching in this site but couldn't find an appropriate answer to the title question: what is the real meaning of the statement "ontogeny recapitulates phylogeny"?
Actually, the statement "ontogeny recapitulates phylogeny" has been largely discredited in modern biology. The stages of embryonic development were thought by Ernst Haeckel to resemble ancestral species in evolution. That is, early embryos appeared to have tails and gills and resembled fish, later embryos resembled quadrupeds, and so on.
However, embryos do NOT actually resemble adults of ANY species. More accurately, one could say that early embryos of different species resemble early embryos of other species, and later stages differ more and more as specialized traits of different species become more obvious. On the early extreme, one fertilized ovum shows very little difference in gross appearance from any other fertilized ovum, and the blastula of different species look very much alike, but as different organs and limbs become better developed, those of different species begin to look more and more different.
This actually only describes SOME of the changes that occur in embryonic development. Other changes are not so simple. One very good book that discusses the strengths and weaknesses in this area is the book "Ontogeny and Phylogeny" by Stephen Jay Gould.