Low poly involves using a low number of polygons in a 3D model. Models include characters, objects, and landscapes. Technically all of the first 3D video games and CG animations such as Vol Libre (1980) were low poly due to the limitations at the time. More recent examples of games include The Very Organised Thief and That Dragon, Cancer, while animated films like The Olympians also utilize the style. 
Low poly games aren’t as demanding as other games. This allows a game to be played by a larger audience as players won’t need high end computers to run it. Having said that there are a group of players that have no interest in games that don't have "good graphics". The approach provides faster load times. Because the game is less demanding and easier to run the developer has the ability to make larger more varied maps.
Secondly, low poly assets are faster to create. There is often no need for texturing. This allows the developers to spend more time focusing on story. Many indie game developers get caught up with the visuals and don't put enough effort into the story. The approach often leads to more work being put into the storytelling.
Finally I feel that the big advantage to low poly games which also applies to all other alternatives to realism is that these approaches do not age. Because they don’t try to rely on looking as realistic as possible they don’t rapidly begin to look outdated as technology continues to improve and games and continue to look more realistic every year.
One of the interesting things about the "low poly" style is how it can look stunning or amateur with very little middle ground between the two extremes. The amateur examples are almost always lacking two things, and surprisingly neither of them is modelling.
The first is lighting. Low poly scenes requires good lighting to look appealing. This can make it difficult to create visually appealing low poly games that have day night cycles or other kinds of dynamic lighting where the designers have less control on the lighting.
The second is called Level of Detail (LOD). Most games use it to improve performance. It involves using low poly models in the distance that are then replaced as the player gets closer with higher poly versions. However low poly games must use Level of Detail to look visually appealing if they have large or open world maps. 
​Take Monet's Poppies below. Notice how in Monet's painting 'Poppies' the grass in the foreground is more detailed than the grass in the background. This reflects how we see things in real life. Subjects in the distance are less detailed. However in a large or open world low poly game if we don't use LOD this will not be so. Objects in the background will be as detailed as those in the foreground and being further away will appear to be more detailed. 
Now take the low poly image below. There's nothing wrong with the lighting or modelling but it lacks the depth it could have if the foreground was more detailed than the background. Similarly the fence and the door models look more detailed than the foreground which goes against what we would perceive in real life.

EDIT: I recently found this talk by Ethan Redd where he explains this, calling it "Rigid Atomic Size".
Using LOD takes a considerable amount of time to set up as it requires the designer to create multiple versions of each model (regular low poly, lower low poly and super low poly). However it is good practice to use LOD in general. The difficulty is when it comes to terrain as large open maps must be separated into chunks and the different low poly chunks will not join correctly sometimes leaving gaping holes. This can be difficult to hide and requires a lot of planning when splitting up the chunks to hide the seams.

Low Poly forest landscape by Varun Bajaj.

Game designers should choose a approach that fits the story or gameplay they wish to create but low poly is a very accessible method of visually conveying a narrative if done correctly. 
I like to think about low poly as digital impressionism. Like the impressionists paintings went against the classical realistic paintings of the time low poly too contrasts the hyper realism that is expected of most games today. Just like impressionist paintings use loose brushwork to leave the viewer with an impression of the subject matter low poly scenes strive to do the same. We are just now reaching the stage of realtime hyper realistic rendering and I hope that after the spectacle wears off people will grow tired of realism and alternative approaches such as low poly and approaches we haven't even discovered yet will gain in popularity.
Be sure to read this article for more information about low poly art and take a look at Prusakov's low poly work for examples of good use of the low poly approach.
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