It was . . . . . . . . proudest moment of my life.

Articles 'a, an and the' are used to determine if a noun is definite or indefinite.
Sometimes an article modifies a noun that is also modified by an adjective.
Here the noun is 'moment' and 'proudest' is an adjective that is modifying the noun. The usual word order is article + adjective + noun, 'proudest moment' is a definite or particular moment that the sentence is taking about, hence 'the' is used.