Why do you put celery in a bloody mary?

There is no chemical reaction that happens. It is mainly a garnish. You might get a tiny bit of celery flavor in the drink from the celery, but I think this is negligible given the other strong flavors in a bloody mary. If you want a celery flavor, add celery salt.

It's meant to be munched on between sips, and permits a "reset" - the cocktail will be fresh upon the tongue the sip after the bite, permitting you to enjoy its flavor and consistency without being oppressed by the cumulative effect of unctuous, spice and salt that is the hallmark of the beverage.

Celery is an astringent and an aromatic - it will cut through the palate-coating tomato juice can leave behind, cool the intensity of the salt and heat, and will interrupt the soup-like consistency with a firm crunch. It's often included in tuna and chicken salad and served with hot wings for the same reason. Other vegetables that tickle our sense of "bitter" in a pleasing way may be similarly substituted - but celery also acts as a handy swizzle stick.

the celery help cleans the taste in your mouth ..helping every sip of your ceaser taste delicious

I've always seen it done as a garnish, and only a few places anymore will keep fresh celery around just for one drink, so it's becoming that much rarer.

To be fair, there are a lot of ways to make a Bloody Mary. A good barman will ask their patron how they like theirs. The two requisites that make it a Bloody Mary are the vodka and the tomato juice, and then a source of salt and sour are typically also considered must-haves, but exactly how you get those in the drink are up to you. Beyond that, any and all of the following are commonly added or omitted based on drinker's preference:

• Clam juice (traditionally a must-have, and a good source of salt, but the name is off-putting in many regions, especially outside New England)
• Pickle juice (for those further inland, this is typically a little more palatable than clam juice to add salt, and also a sour note)
• Olive juice (same idea as pickle juice)
• Jalapeno brine (same but adds a spicier kick)
• Lemon juice
• Lime juice
• Sweet & Sour mix
• Worstershire sauce (yes please; most don't consider it a BM without this)
• Tabasco/hot sauce (ditto)
• Balsamic vinegar
• Kosher salt (some will put it in the drink, others will salt the rim)
• Cracked pepper
• Chili powder (same idea as Tabasco; works best when mixed with the tomato juice ahead of time)
• Sriracha, wasabi, etc (exotic forms of spicy)
• Beer (traditionally a chaser, now it's just as often added directly to the drink)
• Ginger (Heard of it, never tried it; the theory is that since a BM is supposed to settle the stomach after a wild night, the ginger furthers that goal. Also a spicy note)

That's just what goes in the drink. I have seen or heard of all of the following being used as garnish, most of them skewered:

• Celery (traditional)
• Olives (modern cop-out)
• Dill spear or whole dill pickle (common in the Midwest)
• Gherkins/sweet pickles
• Asparagus
• Cocktail onions
• Cherry tomatoes
• Lemon wedge/slice
• Jalapenos (fresh or pickled, whole or slices; obviously common in the Southwest)
• Pepperoncini peppers
• Artichoke hearts
• Pepperoni
• Salami
• Bacon
• Ham (we making a drink or a sandwich here?)
• Slim Jim/beef jerky
• Cheese (various varieties from cubes of pepperjack to string cheese)
• Shrimp
• Tater Tots
• Onion Ring
• Hard boiled egg (say what?)
• Spare rib
• Brisket slice (ok, now it's a combo meal)
• Buffalo wing
• Bacon cheeseburger (yes, an entire bacon cheeseburger on a skewer)

... and of course, someone just had to put them all in one drink:

The celery stick garnish became a staple of the Bloody Mary only after an impatient patron at Chicago’s Pump Room couldn’t wait for his server to bring him a swizzle stick. He took matters into his own hands and snatched a celery stalk from a nearby relish tray.