The following is imprecise, but may help in the visualization. Imagine that Ann does the experiment $2(7700)$ times. We picked this strange number to make the later arithmetic simple.
Then "about" $7700$ times she chooses the box with $3$ orange and $4$ black, and "about" $7700$ times she chooses the box with $5$ orange and $6$ black.
Out of the about $7700$ times she chooses the first box, she gets orange "about" $\frac{3}{7}(7700)$ times, that is, $3300$ times. And out of the roughly $7700$ times she chooses the second box, she gets orange about $\frac{5}{11}(7700)$ times, so roughly $3500$ times.
So she got orange about $6800$ times. And they came from the second box about $3500$ times. So we would expect that the probability the ball came from the second box given that it is orange should be $\dfrac{3500}{6800}$.
It is safer to use the machinery of conditional probabilities. But the above calculation may help give an informal idea of what is going on. In essence we are resricting the sample space to those situations where we got an orange ball.