Can you please explain the reason for having the light of the candle reflect against our finger nails?

The simple Halachic reason is that we want to benefit from the light to say the Bracha of Meorei Haesh. To show that we are benefiting from the light we look at our fingers to tell the difference between the nails and the skin and to show in that way that the light is bright enough to benefit from it. However there are esoteric reasons for that custom as well. The Midrash says that Adam when he sinned he looked at himself and realized he is naked apart from his nails – and that happened on Motzaei Shabbat. Similarly we want to grow like the nails that constantly grow and we do it on Havdala for a good sign for the coming up new week.