That's a tall order of a question. Its got lots of familiar elements, but its really hard to put a finger on what it plays like. Really nothing. It an economics game with a complex action selection mechanism:
1) It has a spatial element, but no real logistics. The routes are set at the beginning of the game and never change. Players effect the map by placing restaurants (their position on the map), adding or upgrading house and advertising. The actual board layout never changes like it does in most games with deliveries (Steam, 18XX, Neuland, etc) and the only way your relative position to a house will change is if someone builds/moves a restaurant closer.
2) The economic engine is wholly in the players hands, but the players never directly exchange anything. Players use advertising to create demand for products, then create the supply to fill that demand. They don't sell that supply to each other, each player has to create what they need. Location and price determine who makes the sales, both of which are determined by actions from the players. In this way, the game is like Steam where players use an actions to both create the opportunities and to fulfill them.
3) The players hire employees each turn. The employees determine what actions the player can take or otherwise modify their actions. Managing a tableau of "actions and privileges" is very reminiscent of many games. It is also extremely strategic, as there is a "tech tree", limited space and a delay in between hiring an employee and getting an effect. There is also a limited number of certain employees, creating a race for them, which is in theme with. . .
4) The milestones that players race for expand the tableau with even more privileges. The early turns don't involve very much in the way of delivering food, but huge decisions are being made as players each attempt to gain an advantage over the others as they carve out a set of powers that will hopefully allow them to out earn the rest of the table. The players aren't building "an engine" like you might in Race to the Galaxy, but they are instead trying to get into a position where they can benefit from other players actions and utilize their actions to harm their competition.
In most economic games, a players position is determined by what they own. They use shares. Players get ahead by manipulating the web of interdependence to hurt other players and help themselves, but they generally do so using the same means. In Food Chain Magnate, your position is determined by what you can do. Your unique combination of employees and milestones is what determines how much money you can make as compared to the other players. It feels completely different from shareholding games because it isn't one.
In the end, the game has elements that remind me of Caylus, Container, Steam, The Great Zimbabwe and In the Year of the Dragon, but it really feels like none of them.
The Great Zimbabwe is my favorite Splotter game. Indonesia is good, but it is a bookkeeping nightmare. You need people who REALLY want to play a mathy game. The other Splotter games are worth playing, but not by me, they focus too heavily on the planning (I enjoy the interaction of leveraging position, which is why I love stock games). I might end up liking Food Chain Magnate better, but one unique thing about TGZ is that its an economic game that people who don't like economic games tend to like. I've introduced it to people who have no interest in anything from something meaty as 18XX to as tame as Power Grid and they have really enjoyed it. FCM may have the same quality, but its more "work" so I have doubts.