It's really complex, and the answer from Shep is a bit imprecise.
Ice at temperatures just below freezing has the remarkable property of not being frozen on the surface. There is a extremely thin layer of liquid water on the surface. How thin? 70 nm at 272 K, but only 10 nm at 262 K. This water layer can act as a lubricant, but with less lubricant the friction is higher.
So, it's not the heat of your finger that causes the liquid layer. It's always there.
The second problem with Shep's answer is the idea of refreezing. It's unclear what he exactly means by that, but a heat wave is not like a wave in water. Heat will diffuse back. You don't get heat ripples.
The icy metal pole mentioned is quite effective in transferring away heat. Metals conduct quite well. The ice cube in comparison doesn't conduct well, and where molten the conductivity is even lower.