The Ionian Boots of Lucidity are a favoured choice on AD casters, who prefer the CDR to the attack speed of the other main offensive AD boot choice, the Berserker's Greaves, and don't tend to get as much CDR from their other items as mages (who for this reason tend to buy Sorcerer's Shoes instead).
Many supports also pick up these boots, as their utility to their team tends to come in a large part from their abilities.
Ionian Boots of Lucidity and a fully stacked Mejai's Soulstealer are the only items that grants an amount of cooldown reduction that isn't a multiple of 10, making them work well with Sorcery. It can be used to reach the CDR cap of 40% without cooldown reduction runes or buying extraneous cooldown reduction.
The Boots of Lucidity were introduced as the item that honors the Ionian Victory in their rematch with Noxus.