Food (たべもの, tabemono) are items that heal and spawn normally as items and from containers. This list does not include Apples from Green Greens, Eggs from Chansey, Birdo's eggs, and the vegetables from the Summit, but these items are all linked to food in that they only recover if food is turned on in the menu. Birdo's eggs and Whispy Woods' apples, for example, will always be hurting items if food is turned off.
There are multiple possible origins where the food could have come from.
In Yoshi's Story, Yoshi would have to eat fruit in order to regain health. When Yoshi eats a total of thirty fruits, the level is complete.
Food that restores health has also appeared in Kirby Super Star as well as many other Kirby series games. There is also a mini-game called Gourmet Race which involves a competition between Kirby and King Dedede on whoever can eat the most food and cross the finish line first. In some later games, food would appear in stages; it was a very abundant, yet low in power, source of health. It is interesting to note, that because the two game series were under the same developer, all food items in Super Smash Bros. Melee were also in the game Kirby Air Ride, reprising their role as common, low-health recovery items.
One more series it could be a reference to is the EarthBound series. Food, such as Hamburgers, serve the purpose of replenishing health in all three games. In Melee and Brawl, food is used to restore small amounts of health. 
Brawl adds an odd technique called Eating While Reclining (倒れアイテム食べ Taore aitemu tabe, Lying and eating items or DownEatU and DownEatD internally in Brawl files, depending on which side the character is lying). If a character lands hard by some food, the player can press A to grab and eat the food without needing to get up first. This technique has no real practical use however, and it can actually directly hinder a knocked down player, as it prevents knocked down characters from using their getup attack if they're near food, limiting their options from this position.
Also in Brawl, there is a special "gulp" sound effect for when characters consume food that happens to be a liquid, like coffee or an orange tonic. When eating vegetables on the Summit, a special "bitty" sound effect can be heard akin to the sounds heard from the original Ice Climber game when a vegetable is collected.
The Food item appears as a trophy in both Super Smash Bros. Melee and Brawl.
 Melee description
These food items will restore your health when you eat them. There are 28 different types of food, and the nutritional value and yumminess quotient differs slightly with each type. Party Balls tend to hold the largest amount of food items. You may be vulnerable when gorging yourself, but you do need to eat to survive.
 Brawl description
Treats that decrease your damage. The nutritional value of each food, and how much damage it heals, is carefully measured. What's better for you, sushi or a donut? Party Balls contain a lot of food, so if you eat it all yourself, you can really lower your damage. There may be times when you laugh and walk away from food, and times when you watch someone else eating and cry.