What does 1/8 watt power rating for resistors actually mean
Easy answer. 1/8 W means 0.125 W.

But what does that mean?

The datasheet will, or should, have details about that. An exhaustive search of two different resistor types in my inventory shows that it means that it can dissipate up to the rated power as long as the ambient temperature is kept below 70 °C.

Note that the resistor will likely elevate the "ambient" temperature.

If you want to operate it at a higher ambient temperature, you have to derate it - that is, use less power. Good resistor datasheet will have curves for this as well, they are pretty simple: