青春の頃 - せいしゅんのころ - the time/days/era of my youth
を
思い出します - おもいだします - to remember/bring up memories of
思い出す is a compound verb that combines the stem form of 思う (to think) with 出す (which has many meanings).
There are many examples of this kind of verb, like
飛び込む - とびこむ - to jump/plunge in
飛び出す - とびだす - to jump/dash out (like the road signs warning you to drive slow because children may run into the street: 飛び出し注意)
投げ出す - なげだす - to throw down/out
貼り付ける - はりつける - to paste/affix something to
This works to elaborate on the manner or direction of the verb, creating a better mental image of the action being performed. I guess the second verb kind of acts like the prepositions that pair with English verbs (into, out of, at, to)