When drawing your dictionary 見送り (みおくり; miokuri) has various meanings, however, the one referring in this article is solely about sending someone off, farewell somebody.
お(御; O), as seen in the title, is a honorific prefix, adding loftiness and beauty (as part of 美化語・雅語; びかご・がご) to the expression, like ‘honorable farewell’.
At train stations, airports, in front of stores or on the streets you may observe this peculiar custom. People bowing and remaining in that very position even after the party has already brought some distance in between or seemingly nobody around at all. For foreigners travelling the country or spending some time living in Japan, this act might appear strange. For Japanese though, this is a natural way to express respect to their departuring guests, clients or business partners. Therefore it can be considered as part of their hospitality to honor somebody’s visit till the very last of moments, as a next encounter may not be foreseeable in the near future. The saying 一期一会 (いちごいちえ) underscores this notion, meaning a once-in-a-lifetime encounter that should be cherished accordingly, including the act of a proper goodbye.
The most occasions I experienced it personally, when leaving a store, after the sales clerk handed over the purchases and escorted me to the doors. The staff would utter an expression of thanks to then make the お辞儀 (おじぎ; bow), until I was no longer in sight. Afterwards the store representative could return and assist another customer in the shop. Among my friends and acquaintances, I have not experienced it that often, but then it was also a more casual setting with gatherings more frequent. What we did when someone left the group after an event, we would normally watch the direction to wave the person goodbye until no longer visible.
Although I didn’t practice the お見送り myself to this extend, I think it is a wonderful custom of respect to the person departing as well as to value the mutual time spent. The world is fast spinning and ephemeral, never knowing, if we are permitted to see each other again under the same circumstances. Then I also believe that last impressions are equally important as first impressions, which is reflected in this custom.