Mostly, they provide a culturally bound (no pun!) sexually attractive shape. They also can prevent the nipples from being visible through outerwear, for those of a more modest turn.
As to preventing sagging, only while wearing them. And the big-busted have to choose between unfashionable shape and nerve damage from the weight on the straps as they cross the shoulders.
Here is the column: Do bras prevent saggy breasts? - The Straight Dope
A friend of mine once proposed a way to find the answer to “do bras prevent sagging?”. Some women, dedicating their bodies to science, will wear a one-sided bra at all times. After many years, the sag quotient of each breast is compared.
Anyone want to volunteer?
And my same friend (RIP) claimed the official name for cleavage was the IBG – Inter-Boobical Groove.
If this is a problem for you, you’re not wearing the right style/size bra for you. Try a molded cup bra with a wide and snug chestband. The chestband is what does the work of holding on to you, and the molded cups keep you snug and contained. The shoulder straps are merely there to keep the tops of the cups from flopping down or sticking out. A well fitted bra, even in my size (40I, ideally, 42DDD or 42G in real life), doesn’t even need shoulder straps to stay up. I’ve been known to slip off my straps and wear 'em in my armpits and still get excellent breast support.
The gold standard of full size bras: http://shop.nordstrom.com/S/2891003?Category=&Search=True&SearchType=guidednav&keyword=bras+in+All+Categories+>+40G%2F+40GG+%2F+40I&origin=searchresults
The almost-as-good-in-some-ways and better-in-others: http://www.catherines.com/pagebuilder/catherines_product_page?pagesize=3&my_nav=intimates&cat=bras&subcat=see%20all&item=0178863&s19b=INTIMATE%20APPAREL%20:%20BRAS&s19c=1
Note the price difference!
Don’t let the molded or “foam-lined” cup scare you off. It’s not a “padded bra”. Bra technology has changed, and molded cups are the best way to get coverage, support, and take that weight off your shoulders. It won’t make you look bigger, it will actually make big women look smaller because you’re more shapely and your breasts don’t bounce and sag as much.
While I do not believe any bra can prevent some sagging, ever watch National Geographic? They have shown some tribes that never were bras, and the older women almost always have extreme amounts of sagging going on, women who might be considered average in a C cup have breasts that are skinny and elongated and hang almost to their navel.
My dad’s favorite joke: What holds a bra up? The gravity of the situation.
(pause)
The gravity of what would happen if it fell down! 
On a more serious note, a bra is a cantilever, IIRC, which is why it doesn’t need the strap to stay up.
My friend’s grandmother was a washer woman in the 20s before bras were popular with the working class. (Initially, they were only for the wealthier women who already could afford the corsets.)
She had long, pendulous breasts. She literally experienced what former Atty. Gen. John Mitchell referred to as getting “her tit caught in a big fat wringer”. Being African American and living in the South, she was not allowed into the White hospitals. So she died.
An unforeseen benefit of the bra is that it prevents some injuries. Shall we call that a “Miracle”?
I don’t have the cite right now (and am not on a computer I’m comfortable searching for it on ;-).
But I distinctly remember reading studies that claimed that wearing bras INCREASED the risk of saggy breasts. They did so by allowing the normal suspensory muscles to atrophy, as the bra did the “heavy lifting,” so to speak.
There are no suspensory muscles. Two structures support the breasts:
The skin brassiere; and
Cooper’s ligaments, which
a. Haven't been carefully studied, but aren't true ligaments; and
b. Predominantly support the functional part of the breast.
The bra-usage-actually-makes-breasts-sag theory has its advocates, but there is little careful science behind it, and certainly none that actually shows muscles atrophying.
While fighting ignorance about breast ptosis, Cecil writes:
Maybe it’s difficult for some. For the rest of us, the Tail Of Spence will find a way to elbow into everyday conversations. 
“Oh, my! How she wags her Tail of Spence! I am entranced.”
A *good *bra is a cantilever. A bad one has, indeed, little more than a decorative band which hangs the cup/breast off the shoulder straps, putting stress on the straps, and therefore shoulders. Most women (I’ve seen estimates northwards of 85% by lingerie salespeople) are not wearing the right size bra, and a too loose good cantilever design can indeed be turned into an ouchy shoulder nightmare if you’re in a bandsize that’s too big.
Most women underestimate their cup size and overestimate their band size. My 60 year old, lifelong A cup wearing mother just got properly fitted into a C In a much smaller band size than she thought she needed.
As far as the not wearing the right size bra, I fit into that camp.
My true bra size is a 30C, but they are impossible to find, so I wear a 32B.
Tail of Spence…Cooper’s Droop
My these people had a LIVELY time with naming parts in the chesticle region.
Don’t forget Montgomery’s Tubercles
http://parenting.ivillage.com/pregnancy/pfirsttri/0,,469b,00.html
I find it ironic that all these wimmins parts have boy’s names. (yeah, I know, we could go on about the old boy’s societies ejumacatin boys only, but work with me here.)
In all the breast remodeling being done these days, I wonder if part of the procedure is reeling in the stretched Cooper’s ligaments? 
After being caught telling a dirty joke in fourth grade, I was made to look up the actual medical term for cleavage, and remember to this day that it is the intermammary sulcus.
But I like yours better.
I’ve lived in an area where women wore no bras (a part of Africa) and it is definately broadly the case that older women have very elongated/ sagging breasts. Now, what I don’t know, is whether that’s because of no bra, or because they average more kids and more nursing. And to be fair I’ve not seen any elderly western women topless, so maybe its just hidden. But it seems plausible/likely to me that a bra does help.
It’s a certain meme among feminists (and repeated in Cecil’s column) that bras are meant to make breasts more appealing to men at women’s inconvenience. Hence, bra burning.
Yet that’s not exactly true. Bras do as much to hide and cover the appearance of youthful breasts as to enhance older ones. It’s really more of an old vs young thing among women than a men vs women thing. Many similar examples of sexism are like that. Eg., the burqa is at least as much about pretty vs ugly and old vs young as it is about sexism.
Sorry, just felt like interjecting that bit of wisdom.
I owe to Spider Robinson one word from my vocabulary: “spathic”. which is defined in the dictionary as “having excellent cleavage.” (It is a term from mineralogy, unfortunately.) 