Roughly speaking, the height of an arithmetic object (number, variety, ...) is a natural measure of its complexity, say in the sense of "how many bits does it take to describe the object." (This is not meant to be rigorous, but you seem to want to know why people use "heights".) One can then ask for heights (complexity measures) that have nice properties, for example, transform nicely (functorially) for maps, i.e., ht(f(object)) is related to some nice function of ht(object). Weil heights and morphisms constitute a nice example of this, and canonical heights on abelian varieties behave even more nicely. For [BSZ], counting elliptic curves by height is more-or-less counting by (1) # of bits to describe the $j$-invariant (the $\overline{\mathbb Q})$ isomorphism class) plus (2) # of bits to describe how twisted the curve is.

Roughly speaking, the height of an arithmetic object (number, variety, ...) is a natural measure of its complexity, say in the sense of "how many bits does it take to describe the object." (This is not meant to be rigorous, but you seem to want to know why people use "heights".) One can then ask for heights (complexity measures) that have nice properties, for example, transform nicely (functorially) for maps, i.e., ht(f(object)) is related to some nice function of ht(object). Weil heights and morphisms constitute a nice example of this, and canonical heights on abelian varieties behave even more nicely. For [BSZ], counting elliptic curves by height is more-or-less counting by (1) # of bits to describe the $j$-invariant (i.e., the $\bar{\mathbb Q}$ isomorphism class of $E$) plus (2) # of bits to describe how twisted the curve is.