Standard translations (e.g., A. D. Godley/Loeb, G. C. Macaulay) convey this idea: Herodotus cannot say exactly what language the Pelasgians spoke, but if one judges from surviving Pelasgian groups (e.g., at Creston above the Tyrrhenians, formerly neighbors of the Dorians in what is now Thessaliotis; those at Plakia and Skylake on the Hellespont who had lived with the Athenians; and other once-Pelasgian places that later changed names), they spoke a barbarian (non-Greek) language.

Therefore, if the whole Pelasgian stock was like this, then the Attic people (being Pelasgian) also changed their language when they became Hellenes. The Crestonians and Plakians still share a language unlike their neighbors, preserving what they brought when they migrated.

Key Greek phrasing includes: the Attic ἔθνος being Pelasgian “ἅμα τῇ μεταβολῇ τῇ ἐς Ἕλληνας καὶ τὴν γλῶσσαν μετέμαθε” (changed/learned the language at the same time as the change into Hellenes). The user’s short quote compresses this argument about Athenians of Pelasgian origin forgetting/changing their original language upon becoming Hellenes.

Standard translations (e.g., Crawley, Smith/Loeb) say roughly: Before the Trojan War, Hellas appears to have undertaken no common action, and it seems the name itself was not yet universal. Before the time of Hellen, son of Deucalion, this appellation did not even exist; the various tribes (especially the Pelasgian) gave their own names to the districts.

When Hellen and his sons grew strong in Phthiotis and were invited as allies by other cities, the peoples gradually acquired the name Hellenes through association, though it took a long time to become general. Homer (later than the Trojan War) is cited as evidence—he does not apply “Hellenes” to all (only Achilles’ followers from Phthiotis) and uses Danaans, Argives, Achaeans instead.

“HERODOTUS (I, 57), about 2500 years ago, wrote:

…it follows that the Athenians, Pelasgians by origin, forgot their first language by becoming Hellenes.”

Thucydides (I, 3) wrote:

“Before the Hellenes, the son of Deucalion, this name does not seem to have existed; each people, and especially the Pelasgians, lent to Hellas a name derived from its own special name.”

Sources

Herodotus, Histories 1.57

Herodotus. The Histories. Translated by A. D. Godley. Loeb Classical Library 117. Cambridge, MA: Harvard University Press, 1920. Book 1, chapter 57.
(Also available via Perseus Digital Library: http://www.perseus.tufts.edu)

Herodotus. The History of Herodotus. Translated by G. C. Macaulay. London: Macmillan, 1890. Book 1, chapter 57.

Thucydides, History of the Peloponnesian War 1.3

Thucydides. History of the Peloponnesian War. Translated by C. F. Smith. Loeb Classical Library 108. Cambridge, MA: Harvard University Press, 1919. Book 1, chapter 3.

Thucydides. The History of the Peloponnesian War. Translated by Richard Crawley. London: J. M. Dent; New York: E. P. Dutton, 1910. Book 1, chapter 3.