Bushi no ichigon comes from the Japanese proverb Bushi no ichigon kintetsu no gotoshi which means "The single word from a warrior is as unbreakable as the bond formed when gold and iron are combined."
Martial artists are supposed to be upright people of principle. If we say we are going to do something then we do it.
This idea of bushi no ichigon is a work in progress for most of us. When I was a student, I used to get into trouble all the time. One of the main reasons why Furuya Sensei would have to scold me was because I "over promised and under produced." Whenever he would ask me to do something, I would always say yes because I wanted him to like me and think favorably of me. What I didn't understand was that Sensei was a "do-er." He liked to get things done and if he asked me to do something it meant that he wanted me to get it done no matter what. It only took a few hundred scoldings to realize this and stop doing it.
This idea of over promising and under producing is something that I see a lot in new students as well. With just a cursory understanding of Aikido and the commitment that it take to master it, they always over estimate themselves. This is not a bad thing per se, but it can lead to misunderstandings, miscommunications and hurt feelings.
It takes a long time to understand one's self and to gain the skill of maintaining healthy boundaries in order to practice bushi no ichigon. When we understand ourselves better and maintain healthy boundaries, we can then fulfill the things that we say that we will do.