Let $(X_n)_{n \in \mathbb{N}}$ be an $L^1$ sequence of random variables that are independently and identically distributed with mean $\mu$. Let $$S_n = \sum_{i=1}^n (X_i - \mathbb{E}[X_i]).$$ The strong law of large numbers gives a pointwise statement about this sequence. Namely, the set of all $\omega$ such that $S_n$ converges strongly to $\mu$, i.e. $$\|\frac1nS_n(\omega)\| \rightarrow 0 \quad \text{as } n \rightarrow \infty$$ has probability 1.
I understand this interpretation and how it differs from the weak law. However, in Klenke's book he gives the strong law of large numbers as the property $$\mathbf{P}\Big[\limsup_{n\rightarrow \infty} \big|\frac{1}{n}S_n\big| = 0 \Big] = 1.$$ How is this definition equivalent to the idea I outlined above? Intuitively, what is the role of $\limsup$ here and why not write it as just $\lim$?
If any sequence has the $\limsup$ of its absolute values equal to 0, then the sequence converges to 0. Simply the $\liminf$ is sandwiched between 0 and the $\limsup$.
I have not read the book you mention. Unless the author explicitly explains, I can only speculate: maybe the $\limsup$ proof of the fact is more direct(?). It is a "weaker" result at face value. However, from the assertion stated above, it is in fact completely equivalent to the "$\lim$" statement.