There was a horse show recently and it was quite, quite warm for the time of year and the fact that the horses were starting to grow their winter coats.
So, why were some of them wearing blankets?
A cooling sheet or "cooler", which may be solid or mesh, is designed to wick sweat and moisture away from a horse's skin and help them dry off after exercise or if they're in a situation where they might break out into a sweat (some horses do so when nervous or when on the trailer). They are often called "anti-sweat" sheets in the UK - they don't stop the horse from sweating, but they make sweat more effective by pulling the water away from the skin.
Some show people also use them to keep the horse clean in the time between grooming and getting into the ring.