Borel-Cantelli Lemma "Corollary" in Royden and Fitzpatrick

I think you're getting a little mixed up (just think about your formulation, intuitively it seems false), but I agree that the conclusion is rather vaguely stated. Let me cite E. Stein's book Real Analysis, in which he poses the Borel-Cantelli lemma as Exercise 16, Chapter 1:

Suppose $\{E_k\}$ is a countable family of measurable subsets of $\mathbb{R}^n$ and that $$\sum_{k=1}^\infty m(E_k)<\infty.$$ Let $$E = \{x\in\mathbb{R}^n: x\in E_k~\text{for infinitely many}~k\} = \limsup_{k\to\infty} E_k.$$ Then $E$ is measurable, and $m(E)=0$.

You can (check this!) write $E = \cap_{n=1}^\infty \cup_{k=n}^\infty E_k$, so if this set is measure zero then almost every $x$ fails to belong to $E$. But if $x$ fails to belong to $E$, then it means precisely that $x$ is contained in at most finitely many $E_k$. This is the desired conclusion, so we're done!

We could state it like that, but no one would know what it means. For its intuitive interpretation, the probabilistic version is very helpful. If you read the $E_k$-s as events and $m(E_k)$ as the probability of the event $E_k$ occurring, then $E$ is the event that infinitely many of the events $E_k$ occur simultaneously. The Borel-Cantelli lemma says that under a suitable decay condition on the probabilities of $E_k$ (namely convergence in the infinite series), the probability of the event $E$ is zero. Believable enough, I think. For its significance, it is an example of a zero-one law: the probability of infinitely many events occurring together is zero. Zero-one laws are of significant interest to probabilists; the page at http://en.wikipedia.org/wiki/Borel-Cantelli_lemma will mention other examples, some related.