This is a really old question but I just recently had to go through some of this stuff so ended up stumbling across it while searching.
OK so there are various basic properties about the tensor product which are useful for getting off the ground:
Given any finite dimensional vector space $V$, we have that $V \otimes V^* \simeq \mathrm{End}(V),$ the space of linear maps from $V$ to itself.
Apply the above fact to the vector space $V \otimes V^*$ and you get that $$(V \otimes V^*) \otimes (V \otimes V^*)^* \simeq \mathrm{End}(V \otimes V^*).$$
On the left hand side we can do some simplification because
- The dual space of a tensor product is the tensor product of the dual spaces; and
- The tensor product of vector spaces is associative and commutative,
So $(V \otimes V^*) \otimes (V \otimes V^*)^*$ is 'unambiguously' (isomorphic to) $V\otimes V \otimes V^* \otimes V^*$, the space of $(2,2)$ tensors. But on the other hand, using the same properties, it is clearly equal to $\mathrm{End}(V) \otimes \mathrm{End}(V)$. And on the right-hand side, we notice that $\mathrm{End}(V \otimes V^*) \simeq \mathrm{End}(\mathrm{End}(V))$, i.e. we have the space of linear maps from matrices to matrices.
But you're asking more about how they act. There basically isn't just one answer; one of the things about all this algebraic stuff is that there are many ways of looking at the same thing and you need to pick a frame that suits what you are actually doing with these objects.
To interpret how a tensor in $\mathrm{End}(V) \otimes \mathrm{End}(V)$ acts as a linear map from $\mathrm{End}(V)$ to $\mathrm{End}(V)$, look at how elementary tensors act. Everything is a square matrix over $V$, i.e. we have $A \otimes B$ and it should act as a linear map on the matrix $M$. To cut a long story short its $M \mapsto AMB^T$.
This again goes via various unravellings of the definitions until you can 'see' it... think of $\mathrm{End}(V)$ as $V \otimes V^*$ again. So then we have an elementary element of $$(V\otimes V^*) \otimes (V^* \otimes V) \simeq \mathrm{End}(V) \otimes \mathrm{End}(V^*)$$ acting on something in $\mathrm{End}(V) \simeq V \otimes V^*$. And the answer to this question will also help - essentially when we use matrices, the rows are covectors and the columns are vectors, so to interpret something like
$$(A \otimes B)(x\otimes y) = A(x) \otimes B(y)$$
in actual matrices, what's happening is that $B$ is acting on the columns of the matrix that $x\otimes y$ represents. And a matrix acting on columns is given by right-multiplying by the transpose.
