Its quite complicated to explain what it actually is. But it basically 'smooths out' packet flow between your PC and the game server, since packets of data from other players are sent periodically and not constantly. If other player's positions were represented by the game 100% accurately, the game would represent their position on the map each time their PC sends a packet of data to the server, so in other words, a 100% true representation of other players by the game, would result in people skipping along the map, instead of running along the ground smoothly. So interpolation basically smooths this out, by predicting where the player would be, before the next packet of data is actually received from his PC, resulting in smooth animations and movement of all players on the map.
The downside to this, is that interpolation actually increases the latency which you experience, and therefore may cause inaccurate hit registry. An example of this manifestation, is when you run behind a wall but still somehow get hit by one or two bullets from the guy who was shooting at you.
Interpolation always existed in BF3. DICE has now just given us control over the amount of interpolation. So lowering the amount of interpolation will give you a more accurate representation of players on the map, but animation smoothness will suffer and you may get stuttering/lagging/jumping/rubberbanding effects if it is set too low. If you set interpolation too high, you get perfectly smooth animations, but your hit reg and accuracy may suffer, as what you are seeing on your screen may not actually be exactly where another player is at that exact moment.
Its a useful feature that DICE has given us, but they have failed to explain it to everyone, and they have failed to explain when and how we should change it, and what the default level always was (before the option was made available in the gameplay settings menu).
Note that my explanation may be incorrect. Its just my understanding of how it works. Perhaps someone else knows abit more.
To change the setting, see the gameplay settings menu in-game.