The definition listed in my glossary is: “Air flow through the vocal tract when producing a sound (e.g. stop, fricative, affricate, nasal, glide, liquid).” But to be more straightforward, manner describes what the parts of your mouth and throat are doing as you say each sound.
Try saying an “ah” sound. This is a vowel and there are no restrictions on the air moving from your throat out of your mouth. Now try saying a “t” sound. Do you notice parts of your mouth that stop air from moving straight through your mouth? Both your tongue and teeth are used to make this sound.
This is just one example of why manner is important for reducing your accent. Once you understand how sounds are produced, it is easier to work on the way that you produce them.