What does Matthew 10:40 mean?

Matthew is coming to the close of his report on Jesus' mission briefing to His core group of twelve hand-picked apostles (Matthew 10:1–4). In the short term, He is sending them out in pairs to the towns and cities of Galilee in northern Israel (Matthew 10:5–8). In the long term, He is sending them out on a lifelong mission to represent Him to the world after His death, resurrection, and return to heaven. Jesus is giving the Twelve great authority. The apostles—named with a Greek term meaning "sent ones"—will teach and heal and cast out demons in His name and in the power of His authority as the Son of God. Jesus has made it painfully clear that this great privilege comes with enormous risk and responsibility.

Because they will go out into the world in the name of Jesus and under His specific authority, those who receive the apostles will, in truth, be receiving Him, as well. Because Jesus also acts on earth under the authority of the Father, whoever receives the apostles will also be receiving God the Father.

What does it mean to "receive" an apostle of Jesus? It includes the idea of gracious hospitality, but it's more than that. Those who received the apostles would also believe and accept their teaching that Jesus was the Messiah, the Son of God. In that way, they would be coming to faith in Christ through the teaching of His official representatives.