Suppose Colin plays A with probability c and B with probability 1-c, while Rose plays A with probability r and B with probability 1-r. Then Rose's payoff is
$2rc+4(1-r)c+10r(1-c)+0(1-r)(1-c)$
Rose's security level is calculated by taking the max-mini payoff: that is, for each value of r, find the minimum payoff over all c, then find the maximum such value.
So, first, we can simplify the above expression:
$2rc+4c-4rc+10r-10rc = 4c+10r-12rc$
As a function of c, this is a line with intercept $10r$ and slope $4-12r$.
Case I: $r = \frac13$, this line is a constant $10r$. Plug $r =\frac13$ in and you get $\frac{10}3$
Case II: $r > \frac13$, then it's negatively sloped, and the minimum occurs at $c = 1$, with a payoff of $4-2r$. Since $r > \frac13$, we have that the payoff is less than $4-2/3 = \frac{10}3$.
Case II: $r < \frac13$, then it's positively sloped, and the minimum occurs at $c = 0$, with a payoff of $10r$. Since $r < \frac13$, the payoff is less than $\frac{10}3$.
In other words, if Rose plays A $\frac13$ of the time, then she gets an average of $\frac{10}3$ regardless of what Colin does. If she plays A less than $\frac13$ of the time, then if Colin plays B, then she gets 10 less than $\frac13$ of the time and 0 the rest of the time, getting her less than $\frac{10}3$. If she plays A more than $\frac13$ of the time, then if Colin plays A, she gets 2 more than $\frac13$ of the time and 4 the rest of the time, getting her less than $\frac{10}3$.
So Rose can guarantee herself $\frac{10}3$ by playing A $\frac13$ of the time, and every other strategy has a possibility of giving her less than that. Thus, $\frac{10}3$ is her security level.
The reason BA should be played at least $\frac13$ of the time is that if BA is played $\frac13$ of the time and AB is played $\frac23$ of the time, then Colin's payoff is $8*\frac13+ 5*\frac23 = \frac83 + \frac{10}3 = \frac{18}3=6$, which is his security level. If BA is played less than $\frac13$ of the time, then his payoff goes below his security level, so he would be better off not cooperating.