I was also confused by this, but I from what I can read the case is that the sounds represented by σ and ζ in Greek are retracted alveolar, which means that their place of articulation is between the sounds written as s and z vs. sh and zh in English. For someone who comes from language where s and z represent dental sounds (e.g. Swedish in my case), the Greek pronunciation sounds even more like sh and zh.
I think modern Greek is kind of loose on their sibilants, so you get a kind of "hissy s" and "hissy z".
I find keeping a loose, relaxed, "lazy" jaw helps me better imitate the Greek voice on here, and the "sh" and "zh" instead of "s" and "z" feel like the natural byproduct of not keeping the mouth shut as tightly.