“And Jesus saith unto him, See thou tell no man; but go thy way, shew thyself to the priest, and offer the gift that Moses commanded, for a testimony unto them.”
And Jesus said to him, See that you say nothing to anyone, but go, show yourself to the priest and offer the gift that Moses commanded, for a proof to them. The silence command and the Mosaic compliance command belong together: the healing is not to be a public spectacle but a private sign. Showing himself to the priest and offering the Leviticus 14 gift is the proper completion of the restoration process within the covenant law — Jesus is not replacing the Torah but fulfilling it, as he said in Matthew 5:17. The for a proof to them (the priests) is the witness that the kingdom has arrived: the priests who examine the healed leper are being confronted with evidence that something unprecedented has occurred.
Church Fathers on Matthew 8:4
After the preaching and teaching, is offered an occasion of working miracles, that by mighty works following, the preceding doctrine might be confirmed.
Because He taught them as one having authority, that He might not thence be supposed to use this method of teaching from ostentation, He does the same in works, as one having power to cure; and therefore, 'When Jesus descended from the mountain, great multitudes followed him.'
While the Lord taught on the mount, the disciples were with Him, for to them it was given to know the secret things of the heavenly doctrine; but now as He came down from the mount the crowds followed Him, who had been altogether unable to ascent into the mount. They that are bowed by the burden of sin cannot climb to the sublime mysteries. But when the Lord came down from the mount, that is, stooped to the infirmity, and helplessness of the rest, in pity to their…