Written Tibetan is unusually hard to romanise because its spelling preserves a much older pronunciation. A syllable can carry a prefixed letter, a superscribed letter, a subscribed letter and one or two suffixes, most of which are no longer sounded, so the written form and the spoken form diverge sharply and a system based on either one loses the other. The romanisations in use before 1959 dealt with this by adding diacritical marks, which worked on a printed page and failed everywhere else: they could not be typed, could not be filed in a card catalogue, and were dropped or mangled in transmission.
Wylie proposed a scheme that uses only the twenty six letters of the roman alphabet, with an apostrophe standing for one Tibetan character, and that maps the letters one to one. Nothing is added and nothing is inferred, so the transliteration can be turned back into Tibetan script without loss, and two scholars writing the same word will produce the same string. It was adopted quickly and became the standard for English language scholarship, and the property that made it succeed on a typewriter is what has kept it in place through card indexes, databases and keyboards.
The timing mattered as much as the design. The scheme appeared in 1959, the year the Dalai Lama left for India, at the start of the period in which Tibetan studies grew rapidly in Europe and North America and a large body of literature had to be catalogued, cited and taught by people entering the field at once. A standard adopted at the beginning of an expansion becomes very hard to displace later, and this one was adopted at exactly that moment.