Determining Length of a walk in Nondeterministic Finite Automata with Lambda Transitions
The length of the walk is the number of edges in the walk. Suppose $p$ is a walk labeled $w$. We can decompose $p$ into $p_1 e_1 p_2 e_2 \cdots p_{|w|} e_{|w|} p_{|w|+1}$, where each $p_i$ consists only of $\lambda$-transitions, and $e_i$ is a transition labeled $w_i$. There is always an equivalent walk in which no transition is taken twice in each $e_i$ (this is what is proved in the paragraph you quote), and so $|p_i| \leq \Lambda$. Therefore the entire walk has length at most $(1 + \Lambda) |w| + \Lambda$.