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When a transition between two states is achieved via any of the symbols $\epsilon,0$, how does the $\epsilon$ effect the transition between the two states?

Does it simply mean that the $0$ is ignored, since it is technically in both states at the same time?

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I suggest you review the definition of epsilon-NFA, especially the definition of when an epsilon-NFA accepts a word.

To answer your question, no, the $0$ is not ignored. Consider an epsilon-NFA with two states, the initial state $q_0$ and the accepting state $q_1$, connected by two edges labeled $\epsilon$ and $0$. This epsilon-NFA accepts the words $\epsilon$ and $0$. If you remove the edge labeled $0$, the epsilon-NFA will accept only $\epsilon$.

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