I don’t really have a source for this, but don’t overthink it. “Friendship” is the official state motto. Which doesn’t really mean anything—state mottos aren’t some guiding ethos everyone in the state has to live by—but they tend to get trotted out for signage and campaigns.
They’re signs asking that you drive in a friendly fashion—watch for pedestrians and cyclists, be careful of your speed, use caution in construction zones, and so on. I’d imagine that this also includes things like avoiding aggressive driving and helping to let others change lanes or merge on/off the freeway.
As with all promotional slogans or etiquette campaigns, I wouldn’t read too much into it, and Texans (I am not one) surely have opinions about their fellow drivers that may be less than friendly. The Texas Department of Transportation is just appealing to drivers to be more friendly (better) drivers.