If fans increase the heat of the air and this DIY AC decreases it, why don't they neutralize eachother? Please try to answer in layman terms, i am only starting to study thermodynamics (And physics)
In my book one of the curiosity parts gives that fans don't actually cool the room but by hitting the air particles they increase their kinetic energy, these high speed particles hit us and hence evaporate our sweat, making us feel cool.
I understand this part, but then recently my friend shown me an homemade air conditioner.
It works by using a table fan and water is siphoned through a tube coiling through the fan. I understand some of the heat that the fan throws out is absorbed by the water which in turn heats up and cools down the air.
But then, why does that actually work? Did the fan not just heat up the air, only for it to be cooled down?
Or in other words, why do they not neutralize eachother? Is the cooling effect provided by this much higher than the fan's heating or is this due to some other concept?
 A: 
these high speed particles hit us and hence evaporate our sweat

Indeed it's evaporation that cools us down, but the particles that hit us are not "high speed" particles in any sense (the warming effect of the fan is very small) and evaporation happens not because of their speed. 
Evaporation always happens when there is a water surface with not enough water vapour next to it (nothing to do with non-water portions of air, only partial water vapour pressure matters). 
To understand how a normal fan cools you down, consider evaporation rate with and without a fan. Without a fan you'll have a lot of water vapour next to your skin because of earlier evaporation, which slows evaporation down. A fan creates air flow that blows a lot of that water vapour away from your skin, therefore making evaporation go faster. 

I understand some of the heat that the fan throws out is absorbed by
  the water which in turn heats up and cools down the air.

By now it should be clear that that is not at all how it works. By injecting water drops into the airflow you create a lot of opportunity for evaporation by creating large water surface with (hopefully) not enough water vapour around it. Thus evaporation happens and air cools down.
The precise meaning of "enough water vapour" is given by a value called equilibrium vapour pressure. (see Vapour pressure of water) When water vapour pressure reaches that level we say that Relative humidity reached 100% and evaporation stops.
With this it's easy to understand the limitations of this DIY AC:


*

*It won't work well if relative humidity in the room is high;

*By evaporation it raises air humidity so it will stop working after a while unless you have a window open to let humid air out and bring dry air in.

A: 
In my book one of the curiosity parts gives that fans don't actually cool the room but by hitting the air particles they increase their kinetic energy, these high speed particles hit us and hence evaporate our sweat, making us feel cool. I understand this part, but then recently my friend shown me an homemade air conditioner.

This is not the case. Fans circulate air around, so they function to move cooler air and displace hotter air. It's unlikely that you would use a fan with the window closed. The effect of the fan blades on the kinetic energy of individual gas molecules is actually very small since gas molecules move very fast.

It works by using a table fan and water is siphoned through a tube coiling through the fan. I understand some of the heat that the fan throws out is absorbed by the water which in turn heats up and cools down the air.

Again not really, some of the heat of the air is absorbed by the tube and the water, and the water takes the heat, but if you leave this in equilibrium the water will eventually reach the temperature of the air and stop absorbing heat. That is why air conditioners need to have a connection inside the room and one outside the room as well, because the heat from the room is expelled to the outside. Similar to why a refrigerator feels warm.
The way a refrigerator or air conditioner expel heat is by expanding and compressing the refrigerant.
A: There are two sides to this issue.
If you imagine perfectly insulated room, with your fan and improvised air conditioner inside the room, and the room is fitted with a very sensitive thermometer.
Issue 1. 
In the closed system you would have, with no fans switched on, the temperature remains constant."Nothing Changes" as our Prof used to say. Turn on fans, whether fitted with water coils or not, you are adding energy to the room in the form of kinetic energy, and the heating of the electrical motor in the fan drawing power from outside the room. The temperature can only go up in the room. With your water coils acting as 'air-con' you will get a localised effect of cooling in front of the fan, but the net temp of the room will still be rising.
Issue 2.
What you are describing is the perceived cooling from the device, compared the the true temperate in the room. If you sprayed cold water on yourself and stood in front of the fan, you would notice a cooling effect from the fan during the time it takes to evaporate all the water, and this could be even more effective than your cooling device. A mentioned in 'issue 1, this is a localised effect in the room. The net temp of the insulated room still goes up due to the energy you have added to the system in the form of electrical power.
