I think in general this is tough, maybe because you can define $Tor$ without ever even saying the word element! Part of the beauty of the thing is that $Tor_1$ can be an obstruction to the preservation of many constructions (all those built out of sequences kernels, cokernels, direct sums and direct products). But you can come up with many physical interpretations this way.
For instance: say I'm in $k[x,y]$ and I want to intersect $I =(x^2, xy)$ and $J = (3x +5, y)$. Is intersecting these two ideals first and extending to $k[x,y,z]$ the same as taking $I (k[x,y,z]) \cap J (k[x,y,z])$? Yes, and you could probably prove it directly for any $I,J$. But you could also write down this construction in terms of kernels and cokernels-- and then just invoke the fact that $k[x,y,z]$ is flat over $k[x,y]$. There's no reason to do it in such a categorical way in this case-- but what if you take $k[x,y]/(x^2-y)$ instead of $k[x,y,z]$? How far are you off now by intersecting before extending instead of after? The answer is roughly "the size of a couple of Tor$_1$'s".
This kind of leaves the higher Tor's a mystery. As anomaly alluded to, when you're working over a local or graded ring $R$ with residue field $k$, there is a unique minimal free resolution $F_i$ of any module $M$ (essentially because of Nakayama's lemma). Then $Tor_i(k, M) = k \otimes F_i$, so that Tor really is just a sequence of vector spaces of the same rank the free resolution.
To see why this can be neat in more geometric setting: take $R = k[x,y,z]$ thought of as the homogeneous coordinate ring of $\mathbb P^2$. We're considering graded modules over this ring, and consequently $Tor_i(M,k)$ is a graded vector space. Let $X$ be a set of six points in $\mathbb P^2$. Sometimes all the points will lie on a conic, but usually not. And via a minimal free resolution, we can check whether or not this is true in terms of the grading on $Tor_1(R/I_X, k)$: If the degree 2 part is nonzero then yes, and otherwise no. Further, if the degree two part is two dimensional then a full pencil of conics contains $X$. This is a bit tautological (we're just considering the degree of the minimal generators of $I_X$). For $Tor_2(R/I_X, k)$ we're playing the "relations between relations" game again, but it's still pretty physical: say that $X$ miraculously lies on $2$ conics $\mathbb V(f_1), \mathbb V(f_2)$. Then are there linear forms $l_1, l_2$ such that $l_1 f_1 = l_2 f_2$? We might expect not, this a lot like assuming that the conics share a common factor-- but maybe it does happen for certain points . Anyways, you can detect whether the conics are close in this sense by looking at the degree $3$ component of $Tor_2(R/I_X, k)$, and so on. Going on like this, anything to do with the betti numbers in the minimal free resolution is translated into a statement about Tors.
It would be cool to have a better interpretation than special cases like these, but I don't know anything good.