Michael Richardson's photographs

Michael Richardson and colleagues in 1997 were instrumental in pointing out the discrepancies between Haeckel's popular diagram and genuine embryos. However, Explore Evolution simply accepts a flawed creationist interpretation of Richardson's work. The claim that the common ancestry of vertebrates requires that embryos be most similar at the earliest embryonic stage was not accepted by Darwin nor Haeckel nor any modern evolutionary or developmental biologist. This is simply a straw man argument that was set up by the misrepresentations of Haeckel's and Darwin's view of embryology and development discussed above.

From Explore Evolution:

Critics of the argument from embryology agree that common descent might be a reasonable inference to draw from the similarity of embryos - if embryos really were similar in their earliest stages of development. But they're not, say most embryologists.
Explore Evolution, p. 68

The inference that earliest embryos of various groups must be similar if they shared a common ancestry is a claim of intelligent design proponents such as Jonathan Wells in Icons of Evolution. Darwin certainly never made such a claim. In his review of Icons of Evolution Jerry Coyne addresses this particular fallacy:

Wells also notes that the earliest vertebrate embryos (mere balls of cells) are often less similar to one another than they are at subsequent stages when they possess more complex features. According to Wells, this counts as evidence against biological evolution, which supposedly predicts that the similarities among groups will be strongest at the very first stages of development. But Darwinism makes no such prediction. Darwin himself noted that embryos must adapt to the conditions of their existence, and the earliest stages of vertebrate embryos show adaptation to widely varying amounts of yolk in their eggs.
Jerry Coyne, (2001) "Creationism by Stealth." Nature, 410, p. 475-476

Michael Richardson and Gerhard Keuck find that Haeckel also did not imply that the earliest stages of embryos must be similar if they share a common ancestor. His concept of caenogenesis, the adaptations to embryonic life which blur recapitulation, also applied to early stages of development, particularly with respect to the size of eggs.

Haeckel was aware of these early differences, and they were included among his caenogenetic exceptions. With regard to egg size for example, he noted that ova of different species look very similar at early stages of maturation (although he did acknowledge that they must show molecular differences; see Haeckel, 1896b: 1, p. 137).
Richardson and Keuck, 2002. "Haeckels ABC of Evolution and Development." Biol. Rev., 77, p. 507

Explore Evolution repeats another false claim from Wells.

This error even crept into the Encyclopedia Britannica, and remains in many modern high school and college biology textbooks.
Explore Evolution, p. 69

This is incorrect. A recent survey of 36 biology textbooks, dating from 1980 to the present and covering high school biology, college introductory biology, advanced college biology, and developmental biology books, found that only 8 of these textbooks mentioned Haeckel or the biogenetic law. Two of these 8 were creationist/ID books (Of Pandas and People, and Biology for Christian Schools from Bob Jones University Press). Of the 6 mainstream textbooks that mentioned Haeckel or the biogenetic law, two are advanced college-level books. In all cases where Haeckel is mentioned (except for the creationist/ID books), the text discussion does not reproduce Haeckel's mistakes.

Explore Evolution emphasizes data from the 1997 paper by Michael Richardson and colleagues that strongly challenged a literal interpretation of Haeckel's diagram.

In 1997, an international team of scientists, led by the embryologist Michael Richardson, compared Haeckel's drawings to photographs of actual embryos at various developmental stages. They found that Haeckel had distorted the evidence at every turn, leading Richardson to tell Science that "it looks like it's turning out to be one of the most famous fakes in biology."
Explore Evolution, p. 69

However, the claim that Haeckel "distorted the evidence at every turn" is untrue. As Michael Richardson and colleagues also point out, there is, in fact, compelling similarity of early embryos which provides strong support for common descent.

Data from embryology are fully consistent with Darwinian evolution. Haeckel’s famous drawings are a Creationist cause célèbre (3). Early versions show young embryos looking virtually identical in different vertebrate species. On a fundamental level, Haeckel was correct: All vertebrates develop a similar body plan (consisting of notochord, body segments, pharyngeal pouches, and so forth). This shared developmental program reflects shared evolutionary history. It also fits with overwhelming recent evidence that development in different animals is controlled by common genetic mechanisms (4)
Richardson, et al., 1998."Haeckel, Embryos and Evolution." Science, 280, p. 983

Furthermore, historians comparing Richardson's images to Haeckel's find other problems. Robert Richards observes:

Richardson and his colleagues selected images from the first edition of Haeckel’s Anthropogenie, which was hastily drawn together from his lectures. The book, though, went through five further editions. With each new edition the text grew fatter as Haeckel deployed more evidence; and the illustration in question expanded the comparison from 8 species of embryo to 20 by the 5th edition (1905). In the subsequent editions, the images grew ever more refined, so that even by the 4th edition (1891), the differences among them became more pronounced. The refinements were a function of more material available and better instrumentation (embryos at the earliest stages are invisible to the naked eye). Had the Science article compared Richardson’s photos with illustrations from Haeckel’s later editions, the argument for fraud would have withered.
Robert J. Richards (2009) "Haeckel's embryos: fraud not proven" Biology and Philosophy 23:147-154.
Richardson vs. Haeckel: Reengineered photographs of embryos with yolk material removed, comparable scaling, and orientationRichardson vs. Haeckel: Reengineered photographs of embryos with yolk material removed, comparable scaling, and orientation

Furthermore, Richardson's own images display disturbing inaccuracies:

several (but not all) of the photographed embryos retain the attached yolk sack and other maternal material; this exaggerates their differences from Haeckel’s images. Haeckel explicitly indicated that he pictured his specimens without yolk, allantois, and amnion (Haeckel 1874, p. 256). The bulge of the salamander is not part of the embryo; rather, it is the yolk sack, as is the case for the fish and the human embryos (though not for the chick and the rabbit, from which the yolk sacks have been removed); moreover the salamander photo is obviously not reduced to the same scale as the others (despite the assertion in the caption for the figure in Science). The chick was photographed in a highly circumflex orientation, which occurs at a somewhat later stage of development than that represented by Haeckel. Again, Haeckel expressly stated that he oriented his embryos all in the same way for ease of comparison. I have used a computer program to remove the yolks in the photographs, scale back the salamander, and straighten out the chick. The result is a bit crude, but one can clearly see that the differences between photograph and illustration are not nearly as great as presented in the Science article. Shorn of yolk, the photographed embryos would not have provided the kind of graphic evidence upon which the Science article was premised.
Robert J. Richards (2009) "Haeckel's embryos: fraud not proven" Biology and Philosophy 23:147-154.

This error is significant, and demolishes Explore Evolution's claim that Haeckel's drawings are inherently unreliable. The inconsistency among Richardson's own images, however, makes them unreliable for use with students, and nothing in this historical dispute offered by Explore Evolution undermines modern evolutionary developmental biology.