It seems Reagan is channeling Æsop's "Fable of the Bundle of Sticks" here—the one about how a single stick breaks easily, but a bundle of sticks is profoundly stronger.

He characterizes the Challenger Disaster as a national tragedy by assuming and insisting that the pain of its aftermath is shared by everyone with everyone. It's not hard to miss:

Nancy and I are pained to the core by the tragedy of the shuttle Challenger. We know we share this pain with all of the people of our country. This is truly a national loss. (3-5)

or:

I want to add that I wish I could talk to every man and woman who works for NASA or who worked on this mission and tell them: "Your dedication and professionalism have moved and impressed us for decades. And we know of your anguish. We share it." (37-39)

He disperses the burden of emotional pain as both a gesture to honor the dead and to unify the country. He implies a message of shared pain combated by shared strength, which is an important idea that suggests sadness can be overcome by courage.

Taking this idea one step further, it implies that the nation is courageous together, not unlike the Challenger Seven, and that courage will allow America to continue moving forward in pursuit of discovery.

Sharing is caring, people, and Ronald Reagan wants you to know it. "In unity is strength." (Source)