Here is just one example, but all throughout my textbook they seem to be saying that it is bad voice leading using aug 2nds. However, I tried to play a lot of the examples and they sound ok on piano. Is this a crazy outdated rule?
The rule comes from an older musical aesthetic in which augmented intervals sounded very strange to the ear. Also, they were considered difficult for singers to sing, and thus forbidden (or, at least, highly discouraged).
Aesthetics have changed, and so has the training and experience of singers. In a voice-leading exercise, an augmented second would be considered wrong, but in real-world music, it wouldn't be a big deal.
In particular (as mentioned by Aaron), the problem is primarily with vocal music in a contrapuntal setting. There was little problem in using augmented seconds in instrumental music. Augmented intervals generally expand and chromatically sharpened intervals tend to move upward (like a temporary leading tone).
Augmented seconds are fine in arpeggiations of a dominant ninth (D-F-G-Ab-B in C minor for example.) I'm not sure about the example as the last two chords are an F#-b with a rising F#-G-A#-B soprano line. The trouble (to me) is that the VI -> V chord has "covered" or "hidden" or "direct" parallel octave movement (GG to F#F#). In addition, there are parallel fifths (GD to F#C#). While these parallels are created with different pairs of voices, the involvement of the bass does make them stand out. The VI-V (parallel chords in root position are often difficult to connect nicely.