The speaker is asking a painful question: was the person who stayed close only doing so to prepare an escape? Being "beside" someone should mean closeness, but here it starts to look like waiting for the right moment to leave. The speaker insists they did nothing wrong, and the confusion in "why, carry me on through?" suggests they still can't make sense of being dragged along and then dropped.
The last couplet turns bitter. The other person treated the relationship like "another show," something to enjoy for entertainment, and when it stopped being fun, they "had to throw in the bin." That image is blunt and hurtful, making the speaker feel disposable, like leftover rubbish. The title, "Another One," adds to this feeling, hinting that the speaker may be just one more in a line of people treated this way. The mood is hurt, puzzled, and quietly resentful, with the short, rhyming lines giving it an almost sing-song bluntness that sharpens the sting.