In obtaining the consistency property of $\widehat\beta$, we applied the Weak Law of Large Numbers. By normalising by $1/n$ we were actually determining what happens to the sample average as $n$ tends to infinity.
We saw that as $n\rightarrow \infty$ the value of $\widehat \beta$ collapses on to the true value $\beta$ – a single point. This is of no use if we want to establish what the distribution looks like as $n\rightarrow \infty$. We require a property that gives a distribution that does not have variance zero as the sample size gets bigger and bigger.
We look at $\widehat \beta$ normalised by an appropriate function of $n$ such that we obtain a distribution that is non-degenerate as $n\rightarrow \infty$. For this we need to move away from the weak law of large numbers that looks at sample averages (sample mean as an approximation of the population mean learned in fundamental statistics), to a theorem that looks at normalised samples averages as $n\rightarrow \infty$.
This brings us to the Central Limit Theorem. If we normalise $\beta$ by $1/ \sqrt n$ instead of $1/n$, asymptotically we obtain a constant non-degenerate distribution.