American electrical items are designed to use 120V. UK sockets supply 230V. If you were to plug American hair straighteners into a UK socket (assuming, of course, that you had a suitable socket adapter), the hair straighteners would heat up to a temperature far higher than intended. This could result in the wiring inside the straighteners burning out or, even more alarming, your hair being burnt by the straighteners.
It is possible to buy socket adapters which incorporate a transformer to drop the voltage to 120V but these can be quite expensive, especially if they're to be used with appliances which require a fairly large current. i.e. it would be cheaper to buy some new straighteners.
Chris