You need to move the converter to the ItemTemplate for the ListPicker. If you use it with the ItemsSource binding then you are telling the framework that you need to convert the value being used for the items collection (this is why the whole collection was being passed to your converter).
EDIT: The converter code needs to be changed as well. The format string should be "{0} days" not "{0 days}"
Here's some xaml that should be close to what you want:
<toolkit:ListPicker x:Name="intervalPicker"
ExpansionMode="ExpansionAllowed"
ItemsSource="{Binding}">
<toolkit:ListPicker.ItemTemplate>
<DataTemplate>
<TextBlock Text="{Binding Path=Days, Converter={StaticResource intervalConverter}"/>
</DataTemplate>
</toolkit:ListPicker.ItemTemplate>
<toolkit:ListPicker.FullModeItemTemplate>
<DataTemplate>
<TextBlock Style={StaticResource PhoneTextTitle2Style}"
Text="{Binding Path=Days, Converter={StaticResource intervalConverter}"/>
</DataTemplate>
</toolkit:ListPicker.FullModeItemTemplate>
</toolkit:ListPicker>