It is because of chlorine.

Chlorine is not the main reason for this.

Chlorine is added to water of swimming pools because it is a disinfectant.

It helps kill germs, thus keeping water germ-free.

In fact,

The main reason for our eyes to turn red is the presence of our bodily fluids like urine and sweat in the pool.

Urine and sweat contain nitrogenous compounds like urea.

Urea reacts with chlorine present in water, forming chloramines.

Now, if we regularly clean the swimming pools, then we can prevent the production of chloramines.

But if the pools are not cleaned, then the chloramines formed can irritate our eyes, thus making them red.

In addition to this, chloramines can evaporate from water.

Inhaling these chloramines for a long period can result in coughing, asthma and other health problems.