The part about not going home when you're done scavenging.
What I said was, "
Over the generations, they might be increasingly selected for spending longer times out of the forest". I thought it would be fairly clear... my bad.
The suggestion is that there would be a selective advantage for those that were more curious, audacious, and quicker across the ground, both empty-handed towards the kill and when returning to the safety of the forest carrying booty. The latter would particularly select for bipedalism.
Once a reasonable level of bipedalism had developed, the potential for greater tool use would be available, including sticks for defence or to scare off other scavengers or even the predators from their kills. This would involve group cooperation, those with sticks enabling others to grab the booty and return to the forest.
This cooperation, together with superior bipedalism and tool use, would allow deeper exploratory incursions into the savannah, foraging.
At some point, these expeditions would find water holes allowing longer stays in the savannah and opportunities for ambush killing rather than scavenging. IOW, a transition to hunting.
Expeditions that required staying out overnight would be far safer when they had developed control of fire, but would be quite possible without it, given trees and shrubs (e.g. near water holes) for protection and cover.
Given the protection offered by fire, it would be possible for the whole group to spend days and nights in the savannah. By this time, the group would be as well, if not better, adapted to the savannah than the forest, so, assuming the savannah provided a better living, they might well find a good niche as hunter-gatherers.
Bear in mind that this would take many generations, and some phases might be far longer than others. The whole progression might be triggered by environmental (e.g. climate) changes that would tip the balance of abundance from forest to savannah - or even result in the loss of forest for savannah, requiring and selecting for an adaptation to the savannah.
This is all speculative and are probably many other potential routes from forest-dwelling to savannah-dwelling, but the point is that one can generate reasonably plausible sequences and reasons for such changes.