Interp is short for "interpolation".

Basically, there is always a time delay between your computer and the remote game server for the data to traverse the internet. That is called "ping". When you click on an enemy's head with the sniper, your computer has to send a packet saying you fired and where you were pointing. The server verifies this, checks the last known position of the enemy, and responds with whether it was a kill or not. Very simplified of course butyou get the idea.

When tf2 was created, many people were gaming on slow and unstable DSL connections. The actual time it took to send and receive data from the server would vary drastically because internet connections were often shared, and sometimes data was lost entirely.

To help reduce the noticable effect of this variability, which would result in jarring, unsteady movement and terrible gameplay, tf2 "interpolates" or intentionally delays data processing on the client side and simply estimates the current state of the game in between update packets from the server. While the game may look smooth and seem like it's live action, What you are seeing on your screen is only what the client THINKS happened in the intervening 100ms or so since the last server update packet happened- or longer if your ping is poor. There's some verification on the server side as well, but the server is basically forced to generally accept the interpolated game state it receives from clients as long as they are within some reasonable range of the server's game state- but the key here is that threshold is largely controlled by the client.

What some cheaters will then do is intentionally crank up the interp delay super high. By forcing the client to guess at the state of the game a very long ways into the future (sometimes a second or longer), the client can claim to have done some crazy kill like landing a spy backstab, and the victim can't do anything about it- because the interp abuser is intentionally delaying server data and replacing it with their own guesstimated data. THAT is why most community servers restrict the interp settings clients are allowed to use- it ensures everyone is on a level playing field when it comes to response time.

That said, with modern cable and fiber internet connections, tf2's method of interp is very outmoded and isn't needed. If you have solid internet, you can actually gain a lot of responsiveness and better game feel by turning interp down to as low as it will go (cl_interp 0 and cl_interp_ratio 1, iirc). There's other specific settings people recommend but you can Google those.