Birkat Hamazon with Wine

Why do some people recite Birkat Hamazon with a cup of wine?

Shalom! 

Thank you for your question. 

Indeed, there is a custom to always recite Birkat Hamazon over a cup of wine. Indeed, some authorities require even one who eats alone to recite Birkat Hamazon over a cup of wine. Most other authorities, however, only require one to recite Birkat Hamazon over a cup of wine when at least three or ten men have eaten together. Nevertheless, there is no true obligation to recite Birkat Hamazon over a cup of wine at any time and common custom is in accordance with this view. Some authorities merely teach that it is praiseworthy to always recite Birkat Hamazon over a cup of wine. The Arizal was of the opinion that Birkat Hamazon should not be recited over a cup of wine when eating alone. 

Some have the custom to recite Birkat Hamazon over a cup of wine on Shabbat and Yom Tov. Among the reasons for this is that, on Shabbat and Yom Tov, people are less rushed and have more time to recite Birkat Hamazon properly. So too, many people endeavor to perform mitzvot more meticulously on these days. Others suggest that the custom of reciting Birkat Hamazon over wine on Shabbat and Yom Tov is in order to be able to recite extra blessings, namely, “borei pri hagafen” after Birkat Hamazon and “al hagefen” after drinking the wine. There is also a view that reciting Birkat Hamazon over a cup of wine on Shabbat and Yom Tov is especially appropriate since one frequently drinks wine during these meals. It is explained that it would be unbecoming to drink wine during the meal and then not do so after the meal in the performance of a mitzva. 

 Tosfot, Pesachim 105b; OC 182 with commentaries.