The given solution is really unclear and hand-wavey.

Instead, just call the expected number of days when facing 3 doors to be x.

In 1/3 of times, it takes zero days.

In 1/3 of times you are delayed by a day and face the same problem, so you expect 1+x days.

In 1/3 of times you are delayed by two days and face the same problem, so you expect 2+x days.

Then,

(1/3)(0) + (1/3)(1+x) + (1/3)(2+x) = x

1+x + 2+ x = 3x

3 = x