Right now I'm here:
(definition of a differential $k$-form on an open set in $\mathbb{R}^n$ as given in baby Rudin, as a function which assigns to each "$k$-surface" a certain number as defined on page 254, definition 10.11)
I'm trying to get here:
"The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form."
I know the definition of a smooth manifold and a Lie group, a little sheaf theory and measure theory, not much else. I don't know the abstract definition of a differential form involving sections of a cotangent bundle, what top forms have to do with measures, or what a cotangent bundle is.
How hard is it to get from (a) to (b)? I really don't want to read a huge textbook to get this down, but I will if I have to.
In short, an $n$-form on an oriented $n$-dimensional manifold allows you to integrate functions over [reasonable] $n$-dimensional subsets. You can define any basis for $T_e^*G$, say $\omega_1(e),\dots,\omega_n(e)$, and turn these into $n$ left-invariant $1$-forms by taking $\omega_i(g) = L_g^*\omega_i(e)$. Then $\omega_1\wedge\dots\wedge\omega_n$ gives a left-invariant $n$-form. If the group is connected, it's unique up to a constant.
Rudin's definition only helps if you're sitting inside $\Bbb R^n$, but if you had a local (orientation-preserving) parametrization of your $n$-dimensional Lie group, you could use the parametrization to "pull back" the $n$-form and get an $n$-form $f dx_1\wedge\dots\wedge dx_n$ on an open subset of $\Bbb R^n$, and then you integrate this just by doing the multiple integral of $f$. [I refer you to my lectures on forms and integration over manifolds linked in my profile if you want to skim through some videos. It's all based on determinants.]