The bolded words refer to an adjective clause .
"that I remember best" is acting as adjective, used to describe "the years." generally, adjective clauses will start with a relative pronoun, making them easier to identify. "that" is a relative pronoun.
It's an adjective because it is a description. The thing it's describing are the years. So the "that I remember best" are describing the significance of the years.
The bolded words "that I remember best" are an adjective clause that modifies the noun "years" in the sentence. It gives additional information by identifying which years are being discussed. The clause begins with the relative pronoun "that," making it clear that it is an adjective clause.
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