I don't understand lemma 1.4.7 page 18 of this introduction to differential geometry (https://www.math.ens.fr/~biquard/idg2008.pdf).
$\varphi_{t}\left ( {x} \right )$ is a solution to the ordinary differential equation $\dot{c}= {X}\left ( c \right )$ with initial condition ${x}$, ${X}$ being a vector field on a manifold ${M}$ .
I don't understand this composition and must be confused. How would this composition work for $\dot{x^{i}}={e^{t}}$
This is a easy consequence of the Picard-Lindölf theorem.
Notice that $t\mapsto\varphi_{t+s}(x)$ is solution of the following Cauchy problem: $$\dot{c}=X(c),c(0)=\varphi_s(x),$$ but by definition $t\mapsto\varphi_t(\varphi_s(x))$ is also a solution, whence the equality.
The claim is true only for time-independent vector-field!
Regarding the example, the equation is not the one you wrote $\dot{x^i}=e^t$, but rather ${\dot{x^i}}=x^i$, since the vector field is given by $X=(x_1,\ldots,x_n)$ and not by $X=(e^t,\ldots,e^t)$ which is not time-independent.
In this case, the flow is given by $\varphi\colon (x,t)\mapsto e^tx$, since $t\mapsto e^tx$ satisfies $\dot{x^i}=x^i$ and has value $x$ at $t=0$. You can check that $\varphi(x,t+s)=e^{t+s}x=e^t(e^sx)=e^t\varphi(x,s)=\varphi(\varphi(s,x),t)$, as claimed.