First I'll explain the relevant sense of for in other contexts, and then I'll explain why the sentence that you found odd makes sense.
"Looking for", "listening for", etc.
Here are some similar uses of for. From your comments, it sounds like you already understand listen for, but I'm providing these just to be sure.
I reached for my keys on the dresser when I woke up, but they weren't there. I spent an hour looking for my keys until I remembered that I left them in my coat pocket last night.
A doctor can apply a stethoscope to a patient's abdomen and listen for gurgling sounds. If there is only silence, the doctor will know that there is a possible blockage. [Source: Take Five Minutes by Ruth Foster (2001), slightly edited.]
Woodpeckers are thought to listen for insects, and owls rely on sound when hunting their prey. [Source: Creating a Bird-Watcher's Journal by Claire Walker Leslie and Charles Roth (1999).]
I've been standing here waiting for a bus for twenty minutes now.
These all indicate directing attention toward finding something decided or imagined in advance, by looking, listening, reaching, etc. The same idea occurs in phrases like "looking for a job".
Suppose that you are sitting indoors and you hear the squeal of tires against asphalt outside, suggesting that the driver of a car on the road outside has just slammed on the brakes to avoid an accident. Most likely, you then listen for the sound of a crash or an impact, because you know that that is likely to happen next, and you are interested to know if an accident occurs and, if so, how bad it is.
"I was going to _____" very often is the first half of a sentence where the second half explains why I did not _____. For example:
I was going to cook breakfast, but we were out of eggs.
I was going to marry him, until I discovered that he was wanted for assault and battery in four states.
I was going to give you a copy of Waiting for Godot, but then I remembered that you hate surrealism.
So, when we hear "I was going to _____," we are primed to next hear a conjunction introducing the part of the sentence that explains why _____ didn't happen. That's what listening for or waiting for meant in WendyG's answer.
If you are wondering why she wrote in the present continuous tense, "I am listening for the 'but' at the end of that," that is a way to narrowly locate her feeling of expectation right at the moment when the sentence "I was going to give you another book" ends. The present continuous tense primarily means action in progress, but commonly takes on variations of that meaning in different contexts. In this context, the primary meaning is altered to "something happening at a very specific moment" as well as "very briefly", i.e. the action is transient. You are right that "The sentence left me listening for the 'but'" means the same thing. Here's another example of this use of the present continuous:
When the car hit the front of the house, I looked up—and the wall is cracking! Luckily it held together.
The shift from past tense to present continuous is a way to make the action seem "more present" to the listener. It's somewhat informal.