if people wake up right in the middle of REM sleep or very shortly thereafter then people report remembering their dreams.

if people wake up a little longer after REM sleep then they don't report remembering their dreams, however.

people who report that they 'don't dream' will report their dreams if they are woken from REM sleep in a sleep lab setting.

if you want to remember your dreams... then i'd start with keeping a journal / diary right beside your bed. as soon as you wake up (are waking up) in that half asleep half awake state... see how much of your dream you can recollect. you might be surprised that you can... then write it down as soon as possible. similarly, while you are going off to sleep... in that half sleep half awake state... as you find yourself drifting off... let your mind wander maybe in the direction of past dreams. if you start to remember then write them down as soon as possible.

the more effort you expend at those kinds of tasks the more you will find yourself remembering.

if that doesn't seem to be working for you then try setting your alarm. maybe half an hour before you are due to wake up or something like that. you want to try and catch yourself during an episode of REM sleep, you see (they don't last long - a matter of seconds even though people report they subjectively seem much longer). we have a number of episodes of them each night...