What's the intuition behind non-integer exponents/powers

Having defined positive integer exponents, if you want the property $$a^m \cdot a^n= a^{m+n}$$ to continue to hold, then you must define $a^0=1$ and $a^{-n} = 1/a^n$ for integer $n$. This takes care of all integers. Then, if you want the property $$a^{mn} = (a^m)^n$$ to continue to hold, you must define $a^{p/q} = \sqrt[q]{a^p}$ (for positive real $a$, and integers $p$ and $q$, $q \ne 0$). This takes care of all rational numbers. And then, if you want the function $a^x$ to be continuous from $\mathbb{R} \to \mathbb{R}$, there is only one such extension.

If we fix $a>0$, $f(x)=a^x$ is continuous on $\mathbb R$. The intuition behind rational exponents is pretty clear, and one extends from the rationals to all reals in this manner.

Yet another intuition follows from the following formula: $$x^{\,y} = e^{\;y\log x}$$ (with appropriate restrictions of course).