What was the purpose of this vestment?
Most likely it had a very practical purpose in the earlier times: to prevent the sacred and precious vestments (chasuble/cope...) from being soiled.
The falda dates back to, at least, the fifteenth century. I would presume it as part of the regalia for a Pope in that period. One thing it did do: When standing at his cathedra or seated on the sedia gestatoria, the falda would flow down, having the effect of making the Pope seem taller than the other dignitaries present!