Not a complete answer but a quick explanation of how I read page 9 in that paper.
- The underlying probabilistic model is that for every line $l\in\mathfrak L'$ you throw a coin that shows head with probability $100/Q$ and tail with probability $(Q-100)/Q$, and then $\mathfrak L''$ is the set of all lines whose coin showed head. In other words, you choose a random subset $\mathfrak L''$ of $\mathfrak L'$ and the probability for choosing a particular set $\mathfrak L_0\subseteq \mathfrak L'$ equals
$$\mathbf{P}[\mathfrak L''=\mathfrak L_0]=\left(\frac{100}{Q}\right)^{|\mathfrak L_0|}\left(1-\frac{100}{Q}\right)^{|\mathfrak L'\setminus\mathfrak L_0|}.$$
- By linearity of expectation the expected cardinality of $\mathfrak L''$ is $\mathbf{E}(|\mathfrak L''|)=\frac{100\alpha N^2}{Q}$, and this implies
$$\mathbf{P}\left[|\mathfrak L''|\leqslant\frac{200\alpha N^2}{Q}\right]\geqslant\frac12.$$
- We are done when we can show that the probability of the event "Every line in $\mathfrak L'$ intersects $N/20$ lines in $\mathfrak L''$" has probability at least $1/2+\varepsilon$ for some positive $\varepsilon$, since then the event "$|\mathfrak L''|\leqslant\frac{200\alpha N^2}{Q}$ and every line in $\mathfrak L'$ intersects $N/20$ lines in $\mathfrak L''$" has probability at least $\varepsilon$.
As I understand it, the intuition is that a typical line in $\mathfrak L'$ intersects a quadratic number of lines in $\mathfrak L'$ so it is highly unlikely that less than $N/20$ of these are chosen for $\mathfrak L''$.
At the moment I don't see how to make that rigorous. I haven't read the rest of the paper, yet, but my impression is that it might be convenient (or even necessary) to replace $\mathfrak L'$ by something slightly smaller, throwing away some rubbish:
- I don't see why $\mathfrak L'$, as it is defined, cannot contain a few exceptional lines that have almost all their intersections with lines outside $\mathfrak L'$, so they intersect less than $N/20$ lines in $\mathfrak L'$. If this is the case, these lines have no chance to intersect $N/20$ lines in $\mathfrak L''$.
- It looks easier to show that with high probability almost every line in $\mathfrak L'$ intersects $\mathfrak L''$ at least $N/20$ times (instead of "every line in $\mathfrak L'$" ). Maybe it is sufficient to replace $\mathfrak L'$ by this big subset.
Let's hope for a better answer by someone who understands what's going on.
