Can somebody help me interpreting the red circled sentences in planer English?
I understand "We view $y_i$ as a realization of a random variable $Y_i$ that can take the values of one and zero" but the next following words, "with probabilities $\pi_i$ and $1-\pi_i$" , make me confused in interpreting the whole sentence.
On
The Bernoulli is the distribution of events that can have only two possible states (0/1; Yes/No; True/False; Heads/Tails; etc.). However, the probability of "0" and "1" do not have to be equal, they only have to sum to 1 (since both options represent the entire universe).
In this example, the woman either is using contraception or not. So the random variable $Y$, representing women's use of contraception a Bernoulli, and an observed woman from the set, the $i^{th}$ woman observed denoted $y_i$ is an observation of a Bernoulli.
Now, we don't know right now what the probability is for woman $Y_i$ to be using contraception, so we'll call it $\pi_i$ ("p"i and "p"robability both starting with "p") and while we don't know what it is, we do know with certainty that the probability of woman $Y_i$ not using contraception is $1-\pi_i$ since that spans the range of possibilities.
On
Saying that $Y$ is a random variable and $y$ is a "realization" is what makes it possible to understand such expressions as "$\Pr(Y=y)$" and "$\Pr(Y\le y)$".
When writing about a density function, the value of $f_Y(y)$ is one number when $y=3$ and another when $y=5$; thus one writes $f_Y(3)$ and $f_Y(5)$, leaving the capital $Y$ intact to identify which random variable it is. The integral $\displaystyle\int_0^1 f_Y(y)\,dy$ is quite a different thing from the integral $\displaystyle \int_0^1 f_Y(Y)\,dy$, since the latter would itself be a random variable, and $\displaystyle \int_0^1 f_Y(Y)\,dY$ is an incoherent expression that cannot refer to anything.
I will try to rephrase, though I don't see a problem in the definition here. You have a random variable $Y_i$. Since it is random it cam take different values with different probabilities. In Bernoulli case it make take only 2 values 0 or 1. It's like you toss a coin. Probability 1/2 it is the head ( realization) and with 1/2 prob is the tail. But result is a random. Now in your example the probabilities are not equal - they are $\pi$ and $(1-\pi)$ because they cover all space of events. The probability might be different too at different trials, so this is why it is defined by $\pi_i$. Formula (3.1) specifies the function of probability for different values of $Y_i$. Indeed when $Y_i=1$, we have $$ \Pr\{Y_i=1\}=\pi_i^1 (1-\pi_i)^0 =\pi_i $$ and $$ \Pr\{Y_i=0\}=\pi_i^0 (1-\pi_i)^1 =1-\pi_i $$