Why does entropy increase with an increase in temperature?
Entropy means the level of disorder in a system. Greater entropy means a less organized system.
To explain further, imagine a beaker filled with pure water. The entropy is low because the water is fairly organized, so the entropy is low.
Now, imagine the water is warmed up. Moderate warming will increase the rate of evaporation. More extensive heating can cause the water to boil.
The water becomes less organized (greater entropy) as water molecules enter into the gas phase because they are much less organized flying around the room than when they were all in the beaker.