A nice account is found in a note to R. Steiner's Die vierte Dimension (1995; translation):
Felix Klein (1845–1925) seems to have been the first mathematician to draw attention to this phenomenon in the early 1870s. According to an account by Zöllner (1878a), Klein spoke with him during a scientific conference on this subject shortly before publishing a treatise (1876) in which he discussed this theme in passing. Klein also reported on their meeting (...) (1926, pp. 169ff). While Klein (1876, p. 478) discusses the subject only in general terms, Hoppe (1879) uses an analytically formulated example to untie concretely a simple three-dimensional knot in four-dimensional space (see also Durège (1880) and Hoppe (1880)).
Klein is also credited by Tait (1877; 1882), Dehn-Heegaard (1907), and himself (1922, p. 67).
Edit. To your additional question: Klein clearly convinced the experts that it wasn't just a conjecture. While he apparently never published his proof, I would bet he used the now-standard argument that one can eliminate one crossing at a time, as in e.g. Seifert-Threlfall (1934, Remark 1), Weitzenböck (1929, §5) or — maybe a first? — the last four paragraphs of Durège (1880).