As Morris points out, “Jesus’ words impose a stiff test.  He gives the man no sign.  The officer has nothing but Jesus’ bare word.  But this is enough.  He rises to the implied demand for faith.  He believes what Jesus says and goes his way.” (p. 291)

Alford says:

The miracle is a notable instance of our Savior not quenching the smoking flax, just as his reproof of the Samaritan woman was of his not breaking the bruised reed. The little spark of faith in the breast of this nobleman is lit up into a clear and enduring flame for the light and comfort of himself and his house.”  (qt’d in PNT)

Jesus answered this man’s request, but not the way he wanted or even expected it to be done. Jesus didn’t do it the man’s way because it would not have helped the man’s faith to grow. A similar example is the centurion who came to Jesus for his servant’s healing. Jesus offered to go with the centurion to his house to heal the man, because the centurion’s faith was great enough that he knew Christ had the authority to just say the word and he would be healed (Matthew 8:8). Jesus does the opposite of that here to grow the man’s faith and the official takes Christ at His word with the result that he gains a much wider view of Who Jesus is. Now he knows that Christ has authority to heal even if He is not present. Also, he who would not have believed had he not seen signs and wonders, now believes just on the words of Christ. Jesus made his faith grow.

Matthew 8:8  The centurion replied, “Lord, I do not deserve to have you come under my roof. But just say the word, and my servant will be healed.

What if the father had not trusted the words of Jesus? Would the boy have lived? Would the household have been saved? It’s critically important to believe what Jesus says to us and then act upon it.