The Launch Room is the small area beneath the arena where each tribute gets to wait out their last few hours before the Games begin:
The hovercraft lands and Cinna and I go back to the ladder, only this time it leads down into a tube underground, into the catacombs that lie beneath the arena. We follow instructions to my destination, a chamber for my preparation. In the Capitol, they call it the Launch Room. In the districts, it's referred to as the Stockyard. The place animals go before slaughter.
Everything is brand-new, I will be the first and only tribute to use this Launch Room.
You're right, the tributes are able to get food and water there:
I struggle to keep my breakfast down as I shower and clean my teeth. [...] "Then there's nothing to do but wait for the call," says Cinna. "Unless you think you could eat any more?"
I turn down food but accept a glass of water that I take tiny sips of as we wait on a couch.
Why can't they smuggle food from the Launch Room into the arena?
In short, because that would be cheating.
No, seriously. Although Katniss's internal monologue tells us later on that "we don't really have any rules to speak of, except don't step off your circle for sixty seconds and the unspoken rule about not eating one another", there are a lot more rules which apply to tributes before entering the arena.
One of them is the rule that they shouldn't fight each other:
"Let the bruise show. The audience will think you've mixed it up with another tribute before you've even made it to the arena."
"That's against the rules," says Peeta.
As we see in this very scene in the Launch Room, another one involves what they're allowed to take into the arena with them.
"It's your district token, right?" I nod and he fastens [the mockingjay pin] on my shirt. "It barely cleared the review board. Some thought the pin could be used as a weapon, giving you an unfair advantage. But eventually, they let it through," says Cinna. "They eliminated a ring from that District One girl, though. If you twisted the gemstone, a spike popped out. Poisoned one. She claimed she had no knowledge the ring transformed and there was no way to prove she did. But she lost her token."
So we already know there's a "review board" to consider the issue of what tributes are allowed to wear as district tokens. Similarly, there must be a rule against smuggling in extra food.
How can this rule be enforced?
There are a couple of safeguards which are likely to stop all but the most determined of cheaters from smuggling food from the Launch Room.
Each tribute has their stylist with them all the time they're there, keeping an eye on them. Although the stylists are friendly with their tributes (at least for District 12), I'm sure they wouldn't allow them to break the rules in this way. For the sake of their own skins, if nothing else: the tributes are being punished enough already by going into the arena, but a rule-breaking stylist could easily be imprisoned and tortured by the Capitol.1
There's a very 1984-esque feel of constant surveillance in the Capitol. The tributes have to be careful what they say, as they suspect they might be being listened to all the time. This surely extends to the Launch Room too: the Capitol will have cameras and microphones there to watch for any wrongdoing.
What would happen if a tribute did smuggle food in?
But at the end of the day, there's no foolproof system for stopping them from doing so. Katniss might have been able to swipe some food when Cinna wasn't looking, and even if the cameras had picked her up, they couldn't exactly stop her - or kill her, or withdraw her from the Games - at that point. So what would have happened if she had?
Well, once the tributes are in the arena, they're more or less at the Gamemakers' mercy. It'd be easy for a cheating tribute to be singled out by some mutt or natural disaster, to come to a quick and grisly end. If the Gamemakers really don't want a particular tribute to win, there's no way they'll be able to.2 It's best to play by the rules unless there's a massive advantage in not doing so. Which brings us to the final question ...
How much would a tribute gain by cheating in this way?
Remember that in order to get away with it, they'll have to fool their stylist, who will have helped them dress and noticed how their clothes fit their body. So they won't be able to fit a great deal of food in their pockets without it being noticeable. At the very most they'd be able to smuggle perhaps one meal's worth. It won't be enough to last them for days. In the long run, it won't make a great deal of difference to their chances.
Spoilers for book 2, Catching Fire
As we find out when Cinna is beaten to death in the Launch Room of the Quarter Quell because he turned Katniss's dress into a symbol of rebellion.
The only reason the District 12 tributes were able to survive in the Quarter Quell, when they were engineered by Snow mainly in order to kill them off, was because the Head Gamemaker was secretly an ally of the rebellion and working to keep them alive, playing Snow for a fool all along.