I found what might be the worst written book on Lie Groups. Ever. Until I find one I like better, I'm going to see if I can persevere through the sludge. I'll write out the theorem word for word and then explain what I can. Hopefully someone can decipher it.

Typically, I use the term "chart" to mean the inverse of what the author uses here. Also, in case it wasn't clear from the title, ##G## is a Lie group.

What does this mean, explicitly using the chart (preferably using ##\varphi_\alpha^{-1}## for his ##\varphi_\alpha##)?

For those of you who might be looking for this in the future, I think I've figured out what the author is trying to say.

Pretend ##G## is an abelian Lie group. Then, the multiplication map ##\mu:G\times G\to G, (a,b)\mapsto ab## is a homomorphism of Lie groups. Consider ##\mu_1=\left.\mu\right|_{\{\mathrm{e}\}\times G}##. Identifying ##\{\mathrm{e}\}\times G\cong G##, we note that ##\mu_1## is the identity. Thus, evaluating the pushforward of ##0\oplus y## by ##\mu_1##, we get ##\mathrm{d}(\mu_1)_{(\mathrm{e},\mathrm{e})}(0\oplus y)=y##. Similarly, for ##\mu_2=\left.\mu\right|_{G\times\{\mathrm{e}\}}##, we get ##\mathrm{d}(\mu_2)_{(\mathrm{e},\mathrm{e})}(x\oplus 0)=x##. By linearity, we get our result: $$\mathrm{d}\mu_{(\mathrm{e},\mathrm{e})}(x\oplus y)=\mathrm{d}\mu_{(\mathrm{e},\mathrm{e})}(x\oplus 0 + 0\oplus y)=x+y.$$

This is used, notably, to prove that ##\exp## is a homomorphism of groups (with an appropriate forgetful functor slapped onto the domain) if and only if the underlying Lie group is abelian.