To delete a place from your My Map:
After opening your My Map, in the left-panel, hover your mouse over the place you want to remove, then click the Delete icon to the right.
To delete a route from your My Map:
After opening your My Map, in the left-panel, hover your mouse over the route you want to remove, then click the Delete icon to the right.
To delete the whole My Map from your collection:
After opening your My Map, in the left-panel, scroll to the very bottom click the red Delete icon.