Assume that you build an automaton for finding occurrences of $bububa$ in the input string. When the automaton reads the input string $bubububa$, then it after reading $bububu$, it must not just return to state $\epsilon$, as otherwise an occurrence of $bububa$ is not detected. Also, jumping back to state $bu$ is would not do the trick, as then again, the $bububa$ starting in letter 3 of the word would be missed. Rather, after seeing a $u$ in state $bubub$, the automaton must transition to $bubu$ in order to no miss the possibility that only the first two letters are to be ignored.
The last line in the definition of $E$ describes how the back transitions in the automaton are found: in order not to miss any occurrence of $bububa$, the state in the automaton jumps back as little as possible.