There are many words where Latin and Germanic begin with s– but the Greek begins with an aspirate (h–). How does this shift come about? They do not seem to be formed in the same part of the mouth at all. (Welsh also often begins with h–!)
This is a common sound change. [h] has no constriction above the larynx, and involves spreading the glottis so that any noise generated is turbulence as the air flows through the glottis. Most of the hearable stuff of [h] is the low-level effect on surrounding sonorants. (On top of that, h might end up partially voiced). In producing [s] especially (other voiceless fricatives do this as well), the glottis needs to be likewise spread, so as to not produce [z], and voiceless fricatives tend to have a very open glottis, compared to stops (the fact of being a stop encourages voicelessness, by rapidly extinguishing any voicing that developed). So every [s] "wants" to become [h]; if the lingual constriction is not as narrow as it is in a proper hissy s, you get an h-like sound, which easily becomes an actual /h/ after some years.