These drinks contain stimulants such as caffiene, or herbal stuff.
When taking in a stimulant, it acts on your sympathetic nervous system
. Your Sympathetic nervous system is your 'flight of fight' system. When it is stimulated, a number of changes occur in the body such as:
Increased heart rate
(which can lead to high blood pressure
(The list goes on an on.......)
If you think about it, cocaine is also a stimulant, and then of course it too would act on the sympathetic nervous system causing the same reactions.
The opposite of the sympathetic nervous system would be the parasympathetic, and reactions are opposite.
I know that may be more than you wanted to know, but you asked "how".