It actually makes some sense when you look at the English sound system. From Wikipedia:
American English has three degrees of labialization: tight rounded (/w/, initial /r/), slight rounded (/ʃ/, /ʒ/, /tʃ/, /dʒ/, coloring /r/), and unrounded, which in vowels is sometimes called 'spread'.
That is, some English sounds such as /ʃ/ are pronounced with rounded lips ("labialized"). On the other hand, as far as I can tell, Korean consonants are labialized only if the following vowel itself is rounded (i.e., ㅗ/ㅘ/ㅙ/ㅚ/ㅜ/ㅝ/ㅞ/ㅟ; I'm not sure about ㅛ/ㅠ).
So, when a Korean speaker hears English words "sake" vs. "shake", they hear two differences:
The place of articulation is different (/s/ vs. /ʃ/), which is roughly analogous to the difference between /세/ and /셰/. (See also this question about "시".)
"shake" is pronounced with rounded lips, which is similar to the difference between /세/ and /쉐/.
In fact, even though modern Hangul does not support the combination, many modern Korean speakers do combine both changes: when they say "셰이크/쉐이크" they actually change both the place of articulation and lip rounding, so that it sounds more similar to the original English word "shake".
(As an aside, if we wanted to faithfully represent it in Hangul, we will probably have to invent a crazy combination like 슈ㅔ이크, but let's not go there...)
In fact, a reverse phenomenon happens with words like "swing" or "Sweden". Because /s/ isn't labialized in English, Korean speakers do not hear these words like "슁" or "쉐덴", because then the ㅅ will have to start with rounded lips! So, instead, we add a "dummy vowel" ㅡ, so that ㅅ is pronounced with flat lips: 스윙 and 스웨덴.