Without written records, how do we know that Polynesian sailors traveled to Hawaii, and even South America?

Here is a quote from the document linked
below:

Abstract Examination of an extensive literature has
revealed conclusive evidence that nearly one hundred species of plants, a majority of
them cultivars, were present in both the Eastern and Western Hemispheres prior to
Columbus' first voyage to the Americas. The evidence comes from archaeology, historical
and linguistic sources, ancient art, and conventional botanical studies. Additionally,
21 species of micro-predators and six other species of fauna were shared by the Old and
New Worlds. The evidence further suggests the desirability of additional study of up to
70 other organisms as probably or possibly bi-hemispheric in pre-Columbian times. This
distribution could not have been due merely to natural transfer mechanisms, nor can it
be explained by early human migrations to the New World via the Bering Strait route.
Well over half the plant transfers consisted of flora of American origin that spread to
Eurasia or Oceania, some at surprisingly early dates. The only plausible explanation for
these findings is that a considerable number of transoceanic voyages in both directions
across both major oceans were completed between the 7th millennium BC and the European
age of discovery. Our growing knowledge of early maritime technology and its
accomplishments gives us confidence that vessels and nautical skills capable of these
long-distance travels were developed by the times indicated. These voyages put a new
complexion on the extensive Old World/New World cultural parallels that have long been
controversial.