Intuition behind the length-scale of the Rational Quadratic Kernel

What is the meaning of the length-scale in a rational quadratic?

$$$$k_{\textrm{RQ}}(t, t') = \sigma^2 \left( 1 + \frac{(t - t')^2}{2 \alpha \ell^2} \right)^{-\alpha} \label{eq:rationalQuadratic}$$$$

I know that the rational quadratic is an infinite sum of squared exponentials with varying length-scales and that alpha sets the 'weighting of the individual length scales'.

I, however, don't understand what the length scale hyper-parameter $$\ell$$ in the form above means. Is this relating to the squared exponential kernels or a completely different hyperparameter?

The bigger $$\ell^2$$ is, the less wiggly your random functions are. This is because you will effectively be blurring together points in a larger window. The squared exponential's analogous parameter, which is often given the same notation, has the same interpretation. So yes, they are related.
I know this isn't a kernel, but if you look at the generalized t distribution for a random variable $$X$$ with parameters $$\mu$$, $$\sigma^2$$ and $$\nu$$, its density is proportional to $$\left(1 + \frac{1}{\nu}\left(\frac{(x - \mu)^2}{\sigma^2} \right) \right)^{-\frac{\nu+1}{2}}$$ and $$\sigma$$ is known as the scale parameter. So this bell curve shape gets wider. If you use this kind of a function as a kernel, this will reinforce the interpretation mentioned above.
Call $$\tau$$ the inverse squared length scale of the squared exponential, $$\exp(- \frac12 \tau \lVert t - t' \rVert^2 )$$.
The rational quadratic kernel follows from taking a gamma distribution over $$\tau$$ with shape parameter $$\alpha$$ and mean $$1/\ell^2$$.
That is, $$\ell$$ in the given parameterization is the mean lengthscale of the "underlying" squared exponential kernel, and $$\alpha$$ is the shape, with $$\alpha \to \infty$$ becoming a squared exponential with lengthscale $$\ell$$, while at $$\alpha = 1$$ it's an exponential distribution.