13:1 Keep on [remain; do not let love fall by the wayside] loving [Gr. philia + adelphos] each other as brothers [treat one another as family members].
13:2 Do not forget [become unmindful] to entertain [treat as a family member; hospitality was important because few inns; do this out of brotherly love rather than out of a sense of duty; use discretion as per 2 John 7-11] strangers [visiting brethren, ministers, believers driven from their homes by persecution; see also Matt. 25:35], for by doing some people [like Abraham in Gen. 18:1-15] have entertained angels [messengers: a guest can turn out to be a messenger of God’s blessings to us] without knowing it.
Note: See Titus 1:8 and Romans 12:13 for additional insight into hospitality. Also read 2 John (warns against entertaining false teachers) and 3 John (offers guidelines for hospitality).
13:3 Remember those [believers] in prison [as a result of their Christian faith; see Matt. 25:36] as if [Golden Rule, Matt. 7:12] you were their fellow prisoners, and those who are mistreated [persecution other than imprisonment] as if you yourselves were suffering [see 1 Cor. 12:26].
13:7  Remember your leaders [possibly leaders who had died, possibly by martyrdom], who spoke the word of God to you.  Consider [observe] the outcome [can refer to the sum total of what one has done in life] of their way of life and  imitate their faith.
13:17 Obey [listen to, follow] your leaders and submit [yield] to their authority. [reasons to submit include…]  They [under-shepherds] keep watch [lose sleep] over you as  men [stewards] who must give an account [to the “great Shepherd”; see 1 Peter 5:1-4]. Obey them so that their work will be a joy, not a burden [groan or anguished sigh], for that would be of no advantage [unprofitable; harmful] to you.
• This exhortation may suggest that church members were not supporting their leaders.
• Read 1 Thessalonians 5:12-13.
• Recommended reading: Spiritual Authority by Watchman Nee.
13:18 Pray for us [indicates the writer of Hebrews was a church leader]. We are sure [persuaded; confident] that we have a clear conscience and desire to live honorably in every way.
H = Heart | Offering hospitality to strangers was a moral imperative and sacred duty in Bible times. God told His people to show their “love for the alien” because they had been “aliens in the land of Egypt” (Lev. 19:34; Deut. 10:19). They knew what it meant to be treated harshly and unjustly. Hospitality is a matter of the heart — of extending to others the kind of care we would want for others to extend to us, especially in difficult circumstances and hostile places.
O = Openness | Offering hospitality requires that we have an openness to those in need. Jesus said that at the judgment He “will say to those on His right … I was a stranger and you invited Me in” (Matt. 25:34-35). And “to those on His left … I was a stranger, and you did not invite Me in” (Matt. 25:41-42). Mother Teresa defined her mission as looking for Jesus in the distressing disguise of the poor and then showing hospitality to those individuals.
S = Sent | People in Old Testament times believed guests were sent to them from God. Abraham showed this attitude when he entertained three strangers who proved to be angels (Gen. 18). The writer of Hebrews admonished his readers to “show hospitality to strangers” as Abraham had done, “for by this some have entertained angels without knowing it” (Heb. 13:2).
P = Philoxenos | The Greek word for hospitality is “philoxenos” — from the words “philo” (love) and “xenos” (stranger or guest). This word means “lover of strangers or guests” and conveys the idea of enjoying being a host. Our English word hospitality is from the Latin word “hostis” which means “enemy or stranger.” The Latin word “hospitem” can refer to a host or guest. Hospitals were originally hostels or hospices, places where pilgrims travelling to the Holy Land found a friendly reception.
T = Table | The table is central to hospitality. Abraham and Sarah prepared a meal for their guests (Gen. 18:2-8). One of the first things a host did was to offer a guest something to drink (Gen. 24:17-18) and then a meal (Gen. 26:30). Lot prepared a meal for his guests (Gen. 19:3). As Christ-followers, we should do the same (Mark 9:41).
A = Acceptance | A remarkable feature of hospitality in Biblical times was receiving an enemy as a guest. Some cultures have a rule that an enemy who has dismounted and touched the rope of a single tent is safe. Enemies enjoyed the same protection as friends (Gen. 19:4-8). I have been a guest in many Muslim homes on my travels. Once, while in the mountains of Pakistan, our host assured us of his protection. “You are my guest,” he said, “and I offer you my food and assure you of my protection.” I felt accepted and safe (Ps. 23:5).
I = Insight | While the Bible encourages hospitality, it also offers a word of caution to those who offer hospitality. In 2 John 10 we read, “If anyone” [a traveling teacher] comes to you and does not bring this teaching [probably refers to the teaching about the incarnation (see verse 7)], do not take him into your house [do not offer shelter and lodging; keep them at a distance because they can endanger the spiritual health of the home and the church] or welcome [in either a private or official way; to give a welcome indicates fellowship and solidarity] him.”
3 John 5 offers this encouragement about opening our homes to God’s servants: “Dear friend [refers to Gaius], you are faithful in what you are doing [Gaius’ service was really a service to Jesus Christ (cf. Matt. 10:40; 25:34-40)] for the brothers [traveling ministers or missionaries; sincere ministers of the gospel as opposed to the false teachers of 2 John; read 1 Cor. 16:6; Titus 3:13], even though they are strangers [same as “brothers” in the first part of this verse; cf. Heb. 13:2] to you.”
T = Time | In Biblical times, a host devoted time to his guests. Guests did not expect privacy and would feel ill-treated or deserted if left alone, even at night. Because people slept with their clothes on, a male host and family members would sleep in the same room as their male guests.
Y = Yes | Offering hospitality seems to be an increasingly difficult thing in our culture. Our calendars are so full that we hardly have time to say yes to guests. Yet, unless we do so our children may not learn the importance of showing hospitality to guests or how to do so. So, let’s heed Paul’s advice in Romans 12:13 and practice, or pursue, hospitality.