Tu B'shvat has its origins not in the Bible, but rather in the Mishna, which was written in the early 3rd century CE. It is primarily an agricultural holiday, as evinced by its other name, the "New Year of Trees".
It was originally a holiday with halakhic (Jewish legal) significance, as it was used to mark the age of a tree for the purpose of harvesting and tithing its fruit - tithes that were given to the priests who served in the Temple and did not own any land.
After the Jewish people were scattered in the Diaspora and were no longer involved primarily in agriculture, Tu B’Shvat became a holiday symbolizing the connection between the Jewish people and the Land of Israel. Dried fruits are symbolically eaten - because fresh fruits from Israel were not available.
Another custom in conjunction with the "New Year of Trees" is the planting, especially by children, of tree saplings. Tu B'shvat is not a holy rest day and businesses are open as usual.