The wave doesn’t move exactly “up” and “down”, it’s a visual representation of higher and lower pressure (in case of sound) and other measurable things such as electromagnetic field intensity in case of light.
Sound is produced by vibrating surfaces. They move back and forth and, consequently, push the air around them back and forth. This creates slightly lower/higher air pressure which then spreads around: if the pressure got higher than ambient in some point, then in one second a point (speed of sound * one second) away would experience the spreading pressure. So, the presure starts oscillating there as well.
For non-sound waves, like light, it works in a similar way: stuff starts changing back and forth at the origin point and then this movement spreads.
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