Prophet Mohammed (peace be upon him) encouraged Muslims to break their fast at the sunset during the month of Ramadan by dates and water. Thus, it is a Sunnah practice to break your fast by eating a date and drinking water.
Date is a good source of instant energy as it contains high nutritional value than can prevent the body to develop mild health problem such as headaches, low blood sugar, and lethargy during the period of Ramadan, when Muslim fast from dawn to dusk. The sweetness of the date takes the edge off hunger too. Thus, prevents a person from overeating after his breakfast. Furthermore, the carbohydrates found in dates also make the fruit a slower digesting food, much better than fried or fatty foods which digest fast and leave one hungry for more!
There is no "significance" of dates in Islam, other than the fact that they are very beneficial to eat. Most of the foods that are mentioned in the Quran are very healthy and dates are one of these foods.
"The pains of labour drove her to the trunk of a date-palm. She [Maryam] said, "Oh if only I had died before this time and was something discarded and forgotten!" A voice called out to her from under her, "Do not grieve! Your Lord has placed a small stream at your feet. Shake the trunk of the palm towards you and fresh, ripe dates will drop down onto you. Eat and drink and delight your eyes..."