Scarcity primarily takes two forms: sexual and nutritional. In alpha-mating species, access to mates is scarce, so it's the individual with the longest horns or the biggest muscles who is the fittest. (See: Red Queen Hypothesis)
In the case of nutritional scarcity, whoever has the best digestion system, or the best foraging skills, or the best hunting skills, survives. These skills are particular to that species's methods for extracting calories from Earth, and so whoever can accomplish the tasks of that species in a way that would be considered the "best" survives.
For human evolution, being the "best" mate or being the "best" calorie-extractor both likely involved being the most intelligent.