how is that expression is generated 
I have no idea that how is red arrowed mammoth term is generated from the yellow arrowed term
Please explain

The author starts by substituting $t$ for $e^x - 1$, hence 
$$ \sin (e^x - 1) = \sin t $$
Now the power series of the sine function is plugged in (yellow arrowed line on the right) 
$$ \sin (e^x - 1) = \sin t = t -\frac{t^3}{3!} + \frac{t^5}{5!} \mp \cdots $$
Now we resubstitute $e^x -1$ for $t$, at the same time we use the power series expansion of the exponential function $$ e^x = 1 + x + \frac{x^2}2 + \frac{x^3}{6} + \cdots $$
or 
$$ e^x  - 1 = x + \frac{x^2}2 + \frac{x^3}{6} + \cdots $$
giving 
$$ \sin (e^x - 1) = \left(x + \frac{x^2}2 + \frac{x^3}{6} + \cdots\right) - \frac 16\left(x + \frac{x^2}2 + \frac{x^3}{6} + \cdots\right)^3 + \frac 1{120}\left(x + \frac{x^2}2 + \frac{x^3}{6} + \cdots\right)^5 \mp \cdots $$
which is the red arrowed expression.
