Hopefully the lovely artist wont mind me answering. Over sleeves are usually pinned on, like she has here. But within 100 years, you'll start seeing paintings with sleeves tied on, with bits of ribbon or cord, and this practice called 'pointing your sleeves' with the strings being the 'points'. But in the 15th century a single straight pin is very common to secure the sleeve.
As to why: for small changes in temperature, such as early morning chill, to midday warmth, it is nice to be able to remove your sleeves or put them back on as needed. They are also nice for visual effect. There are some paintings where a woman might have pretty brocaded sleeves to put on, where she might not be able to afford so much of the same fabric to make a whole dress.