That "ee" sound is known in IPA [0] as /i/. That sound is also the vowel counterpart to the semi-vowel /j/ [1, 2]. Within languages, /j/ has a tendency to palatize sounds. In the case of Korean, this has resulted in /s/ being conditioned to /ɕ/ before /i/, where /ɕ/ is the alveolo-palatal version of /s/. (This sound is technically different than "she", which is /ʃi/.)
This process of moving where a sound is pronounced is actually known as "assimilation", but this particular case of assimilation is via a palatal sound. That's why there's actually two palatalization terms in linguistics -- the one I linked above and this one: https://en.wikipedia.org/wiki/Palatalization_%28phonetics%29
