The seller sends the bitcoins to the transaction service ahead of time. You can see when the transaction is funded, so you know that they have the coins.
At the time of the transaction, the seller sends a text message with their phone to localbitcoins.com (or they log into their account at localbitcoins.com) to release the transaction. localbitcoins.com immediately releases control of those coins to you in your localbitcoins.com wallet, and sends you a text message to your phone to let you know that you now have control. In this way, the buyer doesn't need to have anything with them but the cash, and the seller doesn't need to have anything with them but a cell phone capable of sending a text message. The buyer can confirm before leaving to meet up that localbitcoins.com has the coins.
Beyond the transaction service, additional good advice is to meet up somewhere public, populated, and with decent security. Personally, I like the idea of meeting in a casino if you have one nearby. I've done several transactions at casinos and they've all gone smooth. Basically, you want to make sure that someone doesn't steal the cash that you bring with, without releasing the bitcoins. A thief will be less likely to pull out a gun, or otherwise threaten your life in a populated, secure, public place.
A casino sounds great, I don't have many near me so that wouldn't be that good for me. Also I don't have a car, so can't travel that far unless I pay for a cab but that would be expensive.