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The extra red blood cells make your blood thicker than normal. As a result, blood clots can form more easily. These clots can block blood flow through your arteries and veins, which can cause a heart attack or stroke. Thicker blood also doesn't flow as quickly to your body as normal blood.
People take blood thinners to increase the amount of time it takes blood to form a clot. They don't actually thin the blood. Thick blood would be blood that clots too fast. Oftentimes heart patients have to take blood thinners (anticoagulants such as coumarin) because they already have a tendancy to clot too quickly.