I want to show that for each $a \in \mathbb{R}_{>0}$ the dilation $\vec{p} \to a\vec{p}$ is a Mobius transformation.
I'm told that this can be done by composing the inversions $I_{0,r_1}$ and $I_{0,r_2}$ for suitable $r_1$ and $r_2.$
I don't really know where to start with this. We have only just started looking at Mobius transformations in $\mathbb{R}^2 \cup\{\infty\}$ in the geometry class I'm taking. So far all I've been told about is that it is a finite composition of reflections and inversions.
Apologies for my previous answer - it was incorrect because I misunderstood what you meant by "inversion".
Identify a point $(x,y) \in \mathbb R^2$ with the point $z = x+iy \in \mathbb C$. An inversion with respect to a circle of radius $r$ sends $$ z \mapsto \frac {r^2} {\bar z}.$$
If you compose an inversion with radius $r_1$ with an inversion with radius $r_2$, you get the map $$ z \mapsto \frac {r_2^2}{r_1^2} z. $$
To get a dilation by factor $a \in \mathbb R_{>0}$, choose $r_1$ and $r_2$ so that $$ a = \frac {r_2^2}{r_1^2}.$$