We need to find the solution range for the inequality $|x-1|<2$

We can open the Absolute value by taking two cases

Case 1: x-1 >= 0
=> |x-1| = x-1 (Check this video to know how to deal with Absolute Value problems)
=> x - 1 < 2
=> x < 2 + 1
=> x < 3 ... (1)

Case 2: x-1 < 0
=> |x-1| = -(x-1) = -x + 1 (Check this video to know how to deal with Absolute Value problems)
=> -x + 1 < 2
=> x > 1 - 2
=> x > -1 ...(2)

Combining (1) and (2) we get (Check this video to know how to merge Inequalities)
x > -1 and x < 3
=> -1 < x < 3

So, we will have a line segment connecting -1 to 3 and there will be two empty circles at -1 and 3 indicating that -1 and 3 are not included in the solution.

So, Answer will be A
Hope it helps!

Watch the following video to learn the Basics of Absolute Values