I'm talking about the last few lines in the image. It says : $t_A$ (time measured by A1 as seen by A1) = $t’_A$ (time measured by A1 as seen by Bob)
Why are these two times equal?
The reason given is: "They must agree on the watch settings at the events even though they may disagree in the amount of time that passed between them".
I think, by 'watch settings', they mean the time displayed on the watch.
So, say, an event E occurs and Alice's watch displays time $t_1$ at the start of the event E (in Alice's frame), and $t_2$ at the end of the event. So Alice measures the interval to be $t_2-t_1$. Suppose Bob was also observing Alice's watch. If they both agree on the 'watch settings' at the start and end of the event, then Bob finds that Alice's watch was again showing times $t_1$ and $t_2$ at the start and end of the the event E respectively. So he also concludes that the interval observed by Alice is $t_2-t_1$.
Why would they both agree on the watch settings though? This is against 'relativity of simultaneity'. If 'Alice's watch showing $t_1$' and 'The beginning of the event E' occur simultaneously in Alice's frame, then it does not mean that both will occur simultaneously in Bob's frame as well.