When you refrigerate the bananas they do tend to stay fresher and last longer than leaving them out at room temperature.
If the bananas are not fully ripe then it's best to leave the bananas out on the counter to allow them to ripen faster and then place the bananas in the refrigerator if you're not gonna eat them within a few days to a week.
When you keep bananas in the refrigerator the bananas usually last up to 2 to 3 weeks but even when bananas start turning mushy and brown they are still safe to eat although they don't taste as good.
For me personally I never refrigerate bananas myself because I love snacking on bananas and eat them up way before a week and always replace them with some more fresh bananas from the store.
Bananas are very healthy for you and make a great snack and there's no harm in eating a lot of bananas.
But if you don't eat bananas that often then refrigerating the bananas can help them last longer.