My understanding is that a Shabbat clock is used for turning on electricity during Shabbath , though even though we’re not turning the switch (the clock does that) - the electricity itself is generated by Jews on duty at the electric plant, so how does the clock solve the problem of not using electricity on Shabbat?

Your question is not limited to the use of a shabbat clock but to the use of electricity on shabbat even if it is turned on for all of shabbat before shabbat begins. It is permitted because the power plant runs automatically and there is human intervention only when a problem arises. When a problem does arise, it is halachically permitted to intervene because in large populations ther is a certainty that a correcting a power failure involves Pikuach Nefesh(= preventing a danger to life).