Even in secular nations that have dispensed with religious justification of morality there seems to be a common understanding that people want to be good, not on a rational level but on an emotional one. It's a compulsion, at least it seems that way for many, whichever way they go about it.
My suggestion is that morality is nothing more than a biological function, and the compulsion to do the right thing has no implications beyond that. Moral behaviour satisfies a core component in many of us, in the same way food satisfies our hunger.
Empathy (I'd argue the root of much of secular reasoning) is not inherent in us because of the Golden Rule or other sound arguments, it's just an urge. Most humans have evolved/developed in such a way that psychologically we are predisposed to this, although it obviously competes with self-interest and a sense of justice. It is also an urge that can be dispensed of entirely in some people, whom we usually label as mentally unhinged.
And although we'd like to say we help people because it is right, and often provide reasoning to back it up, fundamentally we are just trying to scratch an itch.