Usually, this is implemented with specialized hardware like a digital signal processor (DSP) that can use minimal power doing this specialized thing while the rest of the system (like the main CPU) is suspended (using minimal power). When the DSP detects the hello phrase, it can wake or interrupt the main CPU to handle the request. This DSP is usually part of the system on chip (SoC) that contains the main CPU.
Stuff like the display, cellular modem, wifi chipset, and main CPU are typically the power hogs.
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