The First Right:
"When you meet him, give him the greeting of peace." For verily the greeting of peace is a cause of love, which results in producing faith (Al-Eemaan), which results in the person entering the Paradise.
The Second Right:
"When he invites you, respond to his invitation." This means that when he invites you with an invitation to some food and drink, then fulfill the request of your brother who has drawn near to you and honored you with the invitation. Respond to his invitation (i.e. accept it), unless you have an excuse.
The Third Right:
His statement, "And when he seeks your advice, advise him." This means that if he seeks consultation with you regarding some action, as to whether he should do it or not, then advise him with that which you would like for yourself. Thus, if the action is something that is beneficial in all aspects, then encourage him to do that, and if it is something harmful, then warn him against it. And if the action contains both benefit and harm, then explain that to him and weigh the benefits against the harms.
The Fourth Right:
"And when he sneezes and praises Allah, then pray for mercy upon him." This is due to the fact that sneezing is a favor from Allah, in the expelling of this congested air that is blocked in certain parts of the body of the human being. Allah makes it easy for this air to have a passage out where it can exit, and thus the sneezing person feels relief. Thus, the Prophet (sallAllahu 'alayhi wa sallam) legislated that the person praise Allah for this favor, and he legislated for his (Muslim) brother to say to him, "May Allah have mercy upon you." He also commanded the person who sneezed to answer his (Muslim) brother by saying to him, "May Allah guide you and set right your affairs."
The Fifth Right:
His statement, "And when he becomes ill, visit him." Visiting the sick is from the rights of the Muslim, and especially for the person who has a highly stressed and emphasized right upon you, like the relative, and the friend, and so forth. It is from the best of the righteous deeds. And whoever visits his Muslim brother, he remains engulfed in the mercy (of Allah), and when he sits with him the mercy (of Allah) covers him.
The Sixth Right:
His statement, "And if he dies, follow him (his funeral)." For verily whoever follows the funeral until the deceased's body is prayed over, then he will receive a Qeeraat of reward. (Translator's note: A Qeeraat is an amount equivalent to the size of the Mountain of Uhud in Madinah.) And if he follows the funeral procession until the body is buried, then he will receive two Qeeraats of reward. And following the funeral procession contains (fulfillment of) a right for Allah, a right for the deceased, and a right for the living relatives of the deceased.