I was just going to get to that, reader Clarissa, thanks! The reason bakers like to brown (clarify) butter is to add more flavor notes to it. As mentioned down below, fats, while they’re great flavor mediums, tend not to add much flavor by themselves. Butter is no exception, though its unique composition does allow us to up the ante a bit by applying heat to it.
So what does that do? Well butter isn’t all fat, as I’ve mentioned before. It contains about 15% water plus up to about 2% protein (casein). Those proteins are what allow butter to “brown.” They’re quite sensitive to heat and start clumping up when they get hot. This is the “froth” that accumulates on the top of a pan of simmering butter: coagulating protein plus water, most of which turns to steam and escapes as the butter cooks.
As the process continues the high heat really takes its toll on those proteins. They begin to break apart into all manner of miscellaneous molecules, brown pigments among them, but also scores — even hundreds — of other whatsits that taste, well…all sorts of ways. The French think that all together they add up to the taste of hazelnuts, which is why they call browned butter “beurre noisette” (hazelnut butter). It’s not a bad description, actually.