Some background as to why they do this can be found in this article. Here's a crucial section:
Saudi authorities destroyed this mausoleum, part of the al-Baqi cemetery in Medina, in early 1926, shortly after taking power in the city in the prior year. In fact, they flattened the entire site, which dated back to the seventh century and is thought to have contained the bodies of some of the prophet Mohammed's early compatriots.
The act "shocked the international Muslim community," Dr. James Noyes, author of The Politics of Iconoclasm, told me.
The Saudis didn't just do this on a whim. They were, and still are, aligned with a religious faction called the Wahhabis — a group of Sunni fundamentalists who, like some Christian denominations, reject any form of worship through religious shrines and icons.
"The attacks on shrines and tombs are a rejection of 'shirk' (the worship of God through shrines)," Noyes explained.
Theologically, Wahhabis and other Islamists trace this back to the story of the golden calf that appears in the Koran and the Bible, in which the Israelites build and pray to an idol, sparking God’s fury. A number of Muslims see the story as a blanket prohibition against the worship of images and shrines altogether.
As the Wahhabis and Saudis consolidated control over what's now Saudi Arabia, they destroyed anything that even hinted at idol worship. "The Arabian peninsula used to have Jewish communities, pagan pre-Islamic tribes, shrines favoured by Shiite and Sufi pilgrims on the Hajj to Mecca and Medina, Ottoman and Egyptian influences, and the Hashemite kingdom," Noyes wrote via email. "All of that is gone."