How do you harvest carrots in creative mode?

You can. But it isn't as easy as survival mode. As UnionHawk said clicking on the crop block will destroy it in creative mode. Instead, you can use flowing water to break the crops. Likewise, you can use pistons to push the crops which will also break the crop block and allow the food to be collected.

Not all foods can be harvested using these methods. For instance, pumpkins and melons are not affected by flowing water. For the purpose of harvesting carrots this should not be an issue, however.

While you cannot harvest by hand, there are other ways to collect your crops.

You can't. The normal way to harvest crops in vanilla is to left click to break them, which drops the harvest similar to mining a block. In creative mode, you always destroy blocks you break. There is no way to harvest crops in creative mode.