When we say perpendicular we mean that your finger should be perpendicular towards the fretboard so that your finger only touches the string you wish to play and not the adjacent strings.

Let's say you're playing the 1st fret on the 2 string but you also want to play the open e string (1st string). If you are not perpendicular on that 2nd string you might end up touching the first string which would mute it. Sometimes you want to mute adjacent string but when you don't, you really need to be perpendicular.

The accidental pull-off you have might come from this. I suppose the pull-offs happens on open strings, right?

Any string that generate sounds means that it has been struck some how. The string need to vibrate to generate sound. That means that there's a slight pull-offs that might come from you taking your fingers off the fretboard at an angle. You need to lift you finger straight up so the string is not bends in any way. This should minimize accidental sound.