Back to my staid etymological persona!
Bridle goes all the way back to Old English, derived from a Germanic root bregdan (pull or twitch), which is also the origin of the word "braid". The -le ending is what is called an "instrumental suffix", meaning "something with which the root verb can be done"; for example a handle is something which you can take in your hand; a girdle is something which girds you. So a bridle, the headgear of a horse including the bit and the reins by which it is controlled, is literally a tool used for pulling.
Bridal, which also goes back to Old English, has a very interesting history. It was originally a noun, literally meaning "bride ale". It used to mean the banquet and other festivities associated with the wedding, when, of course, the Anglo-Saxons would quaff a lot of ale. I guess wedding receptions haven't changed much in 1500 years. But by about 1600, people started to think that "bridal" was an adjective meaning "of a bride". They were influenced by that -al ending, more typically used in adjectives (nuptial, mortal, fatal...) than in nouns.
Remember, using the wrong homophone is something that your spellchecker cannot identify or correct!
PS: some readers have commented that they think the correct expression is "champing at the bit". Please check out my other post about this, Breakfast of... chompions?
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