What causes that to happen and should I remove the stems?
My Anthurium Has Brown Spots on the Leaves
The most likely cause is that the soil is staying too wet. This damages the roots and allows a variety of fungal diseases to invade both roots and leaves. Anthuriums are notoriously difficult plants because their range of adaptability is small -- that means they like exactly what they like, in terms of soil moisture and light, and can't adapt very much to conditions outside the optimum. Try increasing the light a bit, increasing air circulation ( a fan), and allowing the soil to go longer between waterings (but don't let it dry out, they don't like that either.) Here's an article with more tips: https://www.gardeningknowhow.com/houseplants/anthurium/anthurium-care.htm