A way of thinking about it is that a point in the plotted "angular power spectrum" represents an averaged sum over the whole sky of the temperature fluctuation in one part multiplied by the temperature fluctuation in another part separated by some angle from the first - represented on the x-axis of the graph by the multiple moment, where $\theta \simeq 180^\circ/l$ (and is shown along the top x-axis).
This means that if the fluctuations have a characteristic angular scale, this reveals itself as a peak in the power spectrum at the corresponding multiple moment.
Since the units of the temperature fluctuations from the mean are Kelvins, then the power spectrum, which is a sum over the product of fluctuation pairs is in Kelvin$^2$.
The units of $\mu$K$^2$ are used because the fluctuations are of order $10^{-5}$ K and so their product is of order thousands of $\mu$K$^2$.
The $l(l+1)/2\pi$ are just unitless normalising factors to make the power spectrum roughly horizontal.