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?
P.S. If you find the English language fascinating, you might enjoy regular updates about English usage and word origins from Wordlady. Receive every new post delivered right to your inbox! If you are not already subscribed, you can either:
use the subscribe window at the top of this page
(if you are reading this on a mobile device): send me an email with the subject line SUBSCRIBE at email@example.com
Also check out my upcoming Rollicking Story of the English Language courses.
Follow me on twitter: @thewordlady