It is an oft quoted Talmudic teaching. It, by all likelihood, has shaped our attitude to welcoming guests. It may well be why even starter homes in our community, surprising to many a realtor years ago, have to have a guest space. It speaks dramatically to the supreme position that kindness and concern for others holds in our life system.
Loosely rendered, Chazal point out that welcoming guests into our homes takes priority over spending time with Hashem, Himself ("גדולה הכנסת אורחים מהקבלת פני השכינה" - שבת קכז.) Apparently there is no other way to explain the seemingly enormous discourtesy of Avrohom, simply taking leave of Hashem, who had come to visit the ailing elder, in favor of greeting some strangers on the road.
By way of example, this dictum directs shul goers to learn from the graciousness of Avrohom, to welcome all guests with full hearts, even those who may be seated in the exact seat where a friend regularly spends time with Hashem.
Yet when all is said and done, can we justify this patriarchal judgement call? Imagine being wrapped in an intense moment during Ne'ila, "if not now then when?", the pre Ne'ila call of the Chafetz Chaim, ringing in one's mind. You are in another place and some stranger tugs at your sleeve, "psst, sorry to bother you now, I need a place to eat tonight, what's your address?"[1] Would any of us be the students of Avrohom at that moment and generously respond with our address and lose the Ne'ila moment that we have so anticipated? And if we do not see in this teaching any practical instruction, what is the value to which Chazal are pointing?
Perhaps Avrohom was forewarning that at Sinai, Hashem will ask all of us to emulate His "behaviors." Indeed our mandate to become kind, considerate and giving people is the directive to "follow in His ways". Is it not possible that becoming G-d like, if you will, trumps greeting Him and basking in His presence?
It may well be that it is the same idea that is expressed in the Nishmas every Shabbos morning. I have not found a satisfying approach to the end of Nishmas. There we praise Hashem for saving the poor from those who rob him and the weak from those who overpower, instead of thanking Him for reducing the thieves and bullies to begin with. Whereas the answer to the question in our hearts is well beyond our grasp, the words that roll from our lips remind us that the opportunity for us to act in a G-d like fashion is readily at hand. That unquestionably is more valuable than any answer.
Similarly, this impression may give meaning to a phrase in Hallel, "Their idols are silver and gold, made by human hands. They have mouths but cannot speak, eyes but cannot see...their manufacturers should become like them." Rashi and Radak both explain that we are praying for the diminution of idol worshipers - they should become as lifeless and uninfluential as the idols they serve. Yet one could also translate the phrase to teach that those who fashion and worship unresponsiveness will themselves develop ears and eyes that will not hear or see the pain of others, and will not have the hands or legs that try to lighten those loads.
[1] Please see a halachic discussion of this in Chavatzeles Hasharon by Rav Carlebach shlit"a. And "over o'reach" by the Aderes zt"l