Think about it in this way, the two people can come at any time between 2 to 4. We can write these two times down as an ordered pair (x,y), with 2 ≤ x,y ≤ 4. The first coordinate is the time at which the first person arrives; the second coordinate is the time at which the second person arrives. Any such point in that region (call it R) is equally likely. Here, we are treating time as completely continuous, which is not a bad approximation.
Now, find the ordered pairs inside this region for which the coordinates differ by less than 15 minutes (i.e. 0.25 hours). (Hint: inequalities). The two people will meet in this region (call it A). You are looking for the probability that, upon throwing a dart at R, the dart lands in A. Convince yourself that this is given by Area(A)/Area(R).