Those unusual curved window grilles are often called “belly bars” or “potbelly window bars.” Their main purpose is still security: the metal bars help prevent people from climbing through the window. But the outward curve at the bottom creates extra usable space between the window and the grille.
One of the most common reasons for that extra space is flower boxes and planters. With completely flat security bars, there may be little room to place plants on an exterior sill. The curved section allows pots or a window box to extend outward while the window remains protected. This design became especially recognizable on older buildings and in Mediterranean-style architecture, where decorative wrought iron and window plants are common.