In Hales's words, 'motivic measure is to traditional measures what an algebraic variety is to its set of solutions'.

The theory of motivic integration was studied by Kontsevich to prove Batyrev's conjecture that two birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. Obviously, this was rooted in the mirror symmetry in string theory, which asserts that for a given pair of mirror smooth Calabi-Yau varieties $X$ and $Y$, their Hodge numbers have a relation $$h^{p,q}(X) = h^{n-p,q}(Y)$$. Still, it is not necessary that a mirror of a Calabi-Yau variety would be smooth; it can be singular. Hence, the idea was to find the relation between the Hodge numbers of crepant resolutions (which we would not discuss here!).

Given a category of varieties over $k$, we construct the Grothendieck ring of varieties $K_0(var_k)$ generated by the isomorphism classes $[X]$ with relations (and they are called scissors relation), that for any closed subvariety $Y \subseteq X$, $$[X] = [Y] + [X\backslash Y]$$ and there is a ring structure on $K_0(var_k)$ as $$[X]\cdot[Y] = [X \times_k Y]$$ If one wishes to study the motivic ring, then we need to also quotient $K_0(var_k)$ (which is free abelian group) by a congruence relation (which would be about motives). Given $X$ and $Y$ nonsingular projective varieties in $var_k$, we say $[X] = [Y]$ whenever their virtual Chow motives are equal. But this is not part of the definition of $K_0(var_k)$. In essence, we wish to know about the additive invariants, and this is where motivic measure is a good general theory, in fact, universal.

Kontsevich's idea was to define the localization of the Grothendieck ring of varieties $\mathcal{M}_k$. Take the class $\mathbb{L} = [\mathbb{A}^1_k]$ of affine varieties (see this Borisov18). Then $$\mathcal{M}_k = K_0(var_k)[\mathbb{L}^{-1}]$$ and associate to each variety a volume in completion $\tilde{\mathcal{M}}_k$ (we will talk about the arc spaces perhaps some other time). The need of $\tilde{\mathcal{M}}_k$ arises in integration theory.

But for any commutative ring $R$, a motivic measure is a ring homomorphism $$\mu: K_0(var_k) \to R$$ which gives a lot of information about geometrical invariants. Note that the measure does not take values now in only $\mathbb{R}$ as in p-adic integration, but in a ring $R$. In logic, it is also helpful to consider the measures of a formula, not only of a set; see this paper by Hales for more. But I do not understand it to comment now.

If $k=\mathbb{F}_q$, then a measure taking value in $\mathbb{Z}$ is just the counting measure $K_0(var_k) \to \mathbb{Z}$, $[X] \mapsto \#X(\mathbb{F}_q)$. (I am also wondering about its connection to the Weil Zeta function now. Maybe, see motivic zeta function.) The Euler characteristic is also a measure (assume $k=\mathbb{C}$) $\chi_{\mathbb{C}}:K_0(var_{\mathbb{C}}) \to \mathbb{Z}$, $[X] \mapsto \chi_{\mathbb{C}}(X)$ (see Example 1.22 here to see how it is a ring homomorphism). And the Hodge-Deligne polynomial is also a measure encoding mixed Hodge numbers $K_0(var_k) \to \mathbb{Z}[u,v]$ and this was the central theme of birational Calabi-Yau discussion.