With Chebyshev's Theorem applied to the proportion of successes in n trials, we reach to the conclusion that when n approaches infinity, the probability that proportion of successes in n trials taking a value between $\theta-c$ and $\theta+c$ approaches to 1.
$P(\theta-c < \frac{x}{n} < \theta+c) = 1-\frac{\theta(1-\theta)}{nc^2}$
$n\to \infty \Rightarrow 1-\frac{\theta(1-\theta)}{nc^2} \to 1 \Rightarrow P(\theta-c < \frac{x}{n} < \theta+c) \to 1$
It doesn't necessarily mean that the proportion will approach to $\theta$ and therefore the number of successes will approach to $n\theta$, but doesn't the sentence I quoted in title just state that? I ran into it at the Wikipedia page for LLN.
"Almost surely" has a very specific meaning in probability. In this case it means that in the sample space of infinitely many coin flips (which is well-studied — for instance, it's isomorphic to the real numbers, by any of the usual bijections to the reals) under the usual measure applied to this space, the set of all sequences for which the probability doesn't converge (in the usual epsilon-delta sense) is of measure zero.