Where does the meaning of "disguise" in travestire hail from? How does the prefix trans- contribute, partly or wholly, to this meaning? I'm baffled because trans- and vestire in Latin didn't mean "disguise", which appears to have first arisen in Italian.
Travesty and transvestite  are first cousins. Both are compounded of the Latin elements trāns- ‘across’ and vestīre ‘clothe’ (source of English vest, vestment, etc), but they are separate formations. Travesty comes ultimately from Italian travestire ‘change clothes so as to disguise’, formed from the Italian descendants of the Latin elements. This was borrowed into French as travestir ‘ridicule’, and its past participle travesti gave English travesty. Transvestite is a new formation, coined in German in the first decade of the 20th century (although there are a couple of isolated instances of a verb transvest ‘cross-dress’ from the 1650s).
Word Origins (2005 2e) by John Ayto. p 515 Right column.
from Italian travestire "to disguise," from Latin trans "across, beyond; over" (see trans-) + vestire "to clothe" (from PIE *wes- (2) "to clothe," extended form of root *eu- "to dress")