Is there any meaning of tensor contraction?

I don't know if you'll find the following helpful, but let $V$ be a vector space and let $T^i_j$ be a $binom11$-tensor.

Probably you are aware that $V^astotimes V cong operatorname{End}(V) cong mathcal M_n(mathbb R)$, the space of $ntimes n$ real matrices, where $n$ is the dimension of $V$. If you are not, then just convince yourself that the element $phiotimes vin V^astotimes V$ gives you an endomorphism of $V$ by $xmapsto phi(x)v$, and that extending this map by linearity we get an isomorphism.

The contraction $T^i_i$ is simply the trace of this matrix, as you probably also know. For some geometric interpretations of the trace, see this mathoverflow post.

Now let's consider a general $binom mn$-tensor $T^{i_1ldots i_m}_{j_1ldots j_n}$. Let's say for ease of notation that we want to contract $i_1$ with $j_1$. We can view $T^{i_1ldots i_m}_{j_1ldots j_n}$ as a matrix $T^{i_1}_{j_1}$ whose entries are $binom {m-1}{n-1}$-tensors, namely $T^{i_2ldots i_m}_{j_2ldots j_n}$. The contraction of $j_1$ with $j_2$ is the trace of this matrix, which is a $binom {m-1}{n-1}$-tensor as well.

It's all about vector in -> vector out, and generalizations of that. You want a way to act on a vector so that the result is a vector, i.e. not just some set of $N$ quantities, but a set that transforms the right way and therefore can be said to be components of a vector. This is enough to tell you what contraction between a second rank and first rank object is doing. After that you can think about all the other ranks by virtue of outer product. (A higher rank thing can always be considered to be a sum of outer products of vectors and/or one-forms).

For simple physical examples, look at e.g. $D^i = epsilon^i_{, j}, E^j$ and $j^i = sigma^i_{, k} E^k$ for a linear crystalline solid.