The idea behind the Thom isomorphism
$\beta:H^iX \rightarrow H^{n+i}(DE,SE)$
is implicit in the formula
$$\int_{\sigma_{n+i}} \beta(\alpha_i) = \int_{X\cap \sigma_{n+i}} \alpha_i$$
Here $\sigma_{n+i}$ is a singular simplex in $DE$ and
we have written integration for the evaluation of a cochain on a sum of simplices. Also $X\subset DE$ is identified with the zero-section.

The problem with this formula is that it doesn't make sense in full generality: after all $X\cap\sigma_{n+i}$ will not in general be a simplex again. And even if it is, it might be a simplex in many different ways (different parametrizations), so some choices must be made. These problems can be overcome and this is the "miracle" of the Thom isomorphism.

Note that the right hand side also requires an "orientation" of $X\cap\sigma_{n+i}$. This is why you also require an orientation on $E$.

For the Thom class $\tau = \beta(1)$ itself this gives the characterization
$$\langle \tau, \sigma_n\rangle = \sharp ( X \cap \sigma_n )$$
where the intersection points are counted with appropriate signs.
(In $DE$ a generic $n$-simplex has a zero-dimensional intersection with the zero section.)

You might find it helpful to learn something about Thom classes in other (generalized) cohomology theories: in de Rham cohomology and K-theory there are pretty explicit representatives for the respective Thom classes. And nothing beats the elegance of Thom classes in cobordism theories, where you've got a "tautological" Thom class.