I know that it probably has something to do with the fact that our dreams didn't really end at the same time our alarm clock rang, our brains just make us think so for some reason.
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There are two possibilities. One is that we do tend to wake up more at the climax of dreams, and that somehow our dreams can sync up with external input like an alarm clock so that the climax of the dream occurs at the same time as the alarm going off. The second is that this doesn't actually happen; the alarm is just as likely to go off at the climax of the dream as it is at any other point.
If the second situation is true, then why do you end up feeling like you always wake up at the climax of the dream? One explanation is confirmation bias. This is an established bias of human memory, in which we selectively remember events that confirm our ideas, and neglect the events that disconfirm them. In this situation, you have the idea that you always wake up at the climax of dreams, so your memory is biased towards times where you actually did wake up at the climax of dreams. However, there are likely many times where you woke up not during the climax of the dream, but these are less memorable because they don't fit your expected pattern.
I'd reckon this can be due to the chance of you remembering the dream when you wake up. When your dream was mundane (i.e. emotionless), you probably have no reason to remember it - when awake. (see The neuropsychology of REM sleep dreaming for more on the noradrenergic and serotonergic influences on sleeping)
While sleeping, only small time-periods (usually 90 minutes) you're actually dreaming, which can result in short of (speculating) time-gaps between periods of the dream. When you were dreaming of something climatically, you can (just as well) leave your dream state, without you noticing it (Waking up properly: is there a role of thermoregulation in sleep inertia. So when the alarm bells ring, you think you were just in a climaxical (word?) dream, while in real life the clock went on.
I must say, this sounds like an interesting research question, although it might be hard to test, because you need the right kinds of dreams (perhaps TMS can provoke them during sleep). Since dreaming happens during REM sleep, we should are able to tell whether you dreaming (regardless of whether or not you will remember later), based on your brain waves (or using EOG).
Finally, since dreams are pretty familiar to us, they are just as mysterious, so one might think of loads of reasons for it (Dreaming and the brain: from phenomenology to neurophysiology, or, or)... Worth while thinking of it
From my personal interest and research into dreaming over 12 years, I ask you to consider that the climax of a dream may actually be caused by the alarm ringing.
There has been long standing hypothesis that real life content gets incorporated into one's dreams. This phenomenon is rather unpredictable, and an example would be seeing lightning strikes in a dream in response to real life light flashes.
From personal experiences - I've frequently been puzzled by waking up a second or so ahead of perceived alarm going off. I hypothesize that this perception is false, and the alarm has been going off for some time before I awoke. Due to the high intensity of an alarm, I hypothesize that the awakening process somehow influences the intensity of dream imagery.
Regarding the confirmation bias - it can also play a part in your question. Once again, from anecdotal evidence, there are many occasions where the dream just fades into blackness and a person slowly gains the sensation of laying in bed (no alarm).