It's great when fancy poets take a popular phrase and make it sound smart. That's what these "good times" are all about, since what's more important than remembering the happier moments in life? It's also the poem's refrain, title, and main idea, so we might as well give it a closer look here.
Lines 6-8: You know if a speaker repeats something three times in only the first stanza, it has to be important. Even if things were rough for the speaker before those bills got paid, now it's all about good times and not worrying so much about the bad times anymore.
Lines 14-16: By the second stanza, everyone is drunk and having a great time, so those "good times" sound and feel even better now. When we go from having a really hard time to having a better time, little things like bread and liquor tend to taste better than ever.
Lines 17-18: Then by the very end, the speaker (who's much older now) tells all children to remember the good times. So the title (and refrain) also happens to be the big moral of the poem too. Talk about being mighty efficient with only two words.