First of all, in both responses the word "think" is left out and understood. In other words it would be "I am thinking" or "I do think." Do is often added to a verb for emphasis and also stands in for a verb that is omitted. In this case it is doing both.
These are two different forms of present tense. "I do (think)" is simple present tense, which describes a habitual or continuous action, but not necessarily happening right now. Examples would be "I eat cake" or "I do eat cake." Which indicates that when there is cake available I partake of it. But unfortunately I have no cake right now.
That is opposed to the present progressive tense, which is an ongoing action occurring right at this moment: "I am eating cake."
Thus his response to the command, "Think about your kids," is "I am," because his kids are on his mind right now.