Why does 시 sound like "she" and not "see"?

Because ㅅ changes shape before the vowel ㅣ. Everywhere else it is the "s" of sun, but in front of ㅣ it slides back in the mouth and comes out as a "sh". So 시 is "shi" and 사 is "sa". You do not have to do anything special — say it fast and naturally and your mouth will do it on its own.