In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how.
The motivation comes from conjectures on special values of $L$-function. Before the general conjectures were formulated by S.Bloch and K.Kato, there were two settings in which the statement of the conjecture was known and which thus provided the main inspiration: the special value of the Dedekind zeta function of a number field $K$ at $s=0$ and the conjecture of Birch and Swinnerton-Dyer for the $L$-functions of an abelian variety $A$ at $s=1$.
In the first case, the special value of the $L$-function is a rational multiple of a regulator, which is equal to the covolume of the lattice $\mathcal O_{K}^\times$ inside $\mathbb R^{r_1+r_2-1}$ and the denominator of this rational number is the cardinality of the torsion group of $\mathcal O_{K}^{\times}$. In the second case, the regulator involves the lattice of rational points (more precisely, their heights) and the rational value equals the cardinality of the torsion group of rational points of $A$ and of its dual. In both cases, then, the transcendental and rational part involve some lattice.
The works of P.Deligne, A.Beilinson, S.Bloch and K.Kato on special values of $L$-function of motives aimed at generalizing this lattice to general (putative) motives.
But for the Tate motive $X=\mathbb Q(1)$, we have that $H^1(X)$ is $\operatorname{Ext}^1(\mathbb Q(0),\mathbb Q(1))$ (for any realization, that is extending the scalar to $\mathbb R$ and considering Deligne cohomology or to $\mathbb Q_{p}$ for some prime $p$ and considering étale or Galois cohomology). Likewise, Bloch showed in A Note on Height Pairings, Tamagawa Numbers, and the Birch and Swinnerton-Dyer Conjecture (Invent. math. 1980) that all the terms of the BSD conjecture could be expressed in terms of real and $p$-adic realizations of the pairing between $\operatorname{Ext}^1(A,\mathbb Q(1))=\operatorname{Ext}^1(\mathbb Q(0),A^*(1))$ and $A$.
Based on these evidence, it seems reasonable to conjecture that $\operatorname{Ext}^1(\mathbb Q(0),M)$ is the group playing for any motive the role rational points play for abelian varieties (extension in the category of mixed motives, whatever that means). This is further confirmed by the fact that Beilinson constructed (conjecturally) a regulator map with source in this space after extension of scalars to $\mathbb R$. Finally, a complete formalism including $p$-adic places was developed by Jean-Marc Fontaine and Bernadette Perrin-Riou (see for instance Valeurs spéciales des fonctions $L$ des motifs Fontaine (1992)).
Note that it would be more accurate to say that this group actually corresponds to the rational points of the dual motive (as this is what happens for abelian varieties if I didn't mess up my normalizations) and that this group is an analogue of rational points only in the limited sense of conjectures on special values of $L$-functions (you could equally validly call it the group of units of a motive, adapting the vocabulary of number fields rather than of abelian varieties). For other properties of rational points, especially diophantine ones, there is no reason to expect that this group will be interesting, which I guess explains M.Kim's remark. To see clearly the discrepancy, you can remark that the Ext group is supposed to be a Chow group when $M$ is pure of weight $-1$ so involves rational cycles, not points, when the motive comes from a variety of large dimension.
I am a bit puzzled by the "even" part of the remark as the category of mixed motives, if it exists and if all the conjectures about it are true (a rather bold requirement, admittedly), is of cohomological dimension 1, so no higher non-trivial $\operatorname{Ext}$ groups are believed to exist.
