An alternative wording for the same idea:

If you are at picture 1, you have 100% chance of selecting it as the current winner.

If you are at picture 2, you have 1/2 chance of selecting it as the current winner, or 1/2 chance of keeping the previous fairly selected winner.

At picture 3, 1/3 chance of picking it, or 2/3 chance of retaining the previous fairly-selected winner. There are two of them, so 1/3 chance of each.

At picture n, you have a 1/n chance of picking it, or an (n-1)/n chance of retaining the previous fairly-selected winner. There are n-1 previous pictures, so all of them have had 1/n chance of being picked.

At every single step, there is the invariant of all pictures being considered that far having had an equal chance of being selected, and the next step always retains the invariant.