Decoherence simplified
Consider the following qubit-environment interaction:
\begin{aligned}
|0\rangle|e\rangle &\longmapsto |0\rangle|e_{00}\rangle
\\|1\rangle|e\rangle &\longmapsto |1\rangle|e_{11}\rangle
\end{aligned}
where |e\rangle, |e_{00}\rangle, and |e_{11}\rangle are the states of the environment, which not need to be orthogonal.
Let |\psi\rangle = \alpha|0\rangle + \beta|1\rangle be the initial state of the qubit.
The environment is essentially trying to measure the qubit and, as the result, the two get entangled:
\Big( \alpha|0\rangle + \beta|1\rangle \Big) |e\rangle
\longmapsto
\alpha |0\rangle|e_{00}\rangle + \beta |1\rangle |e_{11}\rangle.
This state can also be written as
\begin{aligned}
\Big( \alpha|0\rangle + \beta|1\rangle \Big) |e\rangle
\longmapsto
& \Big( \alpha|0\rangle + \beta|1\rangle \Big) \frac{|e_{00}\rangle+|e_{11}\rangle}{2}
\\+& \Big( \alpha|0\rangle - \beta|1\rangle \Big) \frac{|e_{00}\rangle-|e_{11}\rangle}{2}.
\end{aligned}
or as
|\psi\rangle|e\rangle
\longmapsto
\mathbf{1}|\psi\rangle|e_{\mathbf{1}}\rangle + Z|\psi\rangle|e_Z\rangle,
where |e_{\mathbf{1}}\rangle = \frac12(|e_{00}\rangle + |e_{11}\rangle) and |e_Z\rangle = \frac12(|e_{00}\rangle - |e_{11}\rangle).
We may interpret this expression by saying that two things can happen to the qubit: nothing \mathbf{1} (first term), or phase-flip Z (second term).
This, however, should not be taken literally unless the states of the environment, |e_{\mathbf{1}}\rangle and |e_Z\rangle, are orthogonal.
This process is what we refer to as decoherence.