How does energy conversion happen (chemical to mechanical?) in moving body muscles?
The conversion from chemical to mechanical energy is in general done by molecular motors which perform a power-stroke, using the hydrolysis of adenosine triphosphate (ATP) or more rarely of GTP to power a change of their conformation while bound to cytoskeletal elements.

Molecular motors include myosins, which is the molecular family powering muscles in interaction with actin filaments but also other cells e.g. for their motility or for morphogenesis. animation  of muscle contraction

There are also kinesins and dyneins, not used for muscle function, but which perform e.g. transport of "cargo" organelles along microtubules.

An alternative way to produce mechanical energy is via ion pumps, as is the case in flagellar rotors.