Like in the title said why can they use magic in StarCraft? Shouldn't it be the science fiction genre, and if they can why can't the Terran use it ?
Clarke's Third Law reads:
Any sufficiently advanced technology is indistinguishable from magic.
What the protoss do may seem as magic to us, but to them it is simply highly advanced technology combined with their innate psionic abilities. From the protoss fact file on Battle.net:
For thousands of years, the ancient and mysterious protoss have used their alien technology and psionic power to maintain order and peace in their part of the galaxy.
With their highly advanced technology and potent psionic abilities, the protoss once considered themselves the most powerful species in the galaxy.
Individual protoss warriors are unsurpassed in their skills, greatly enhanced by the combination of technology and formidable psionic strength.
What might seem as magic is their psionic abilities, but when you look at other franchises, it is not so different from, say, Invisible Woman's shields, or Jean Grey's telekinesis and telepathy.
From the same fact file:
The protoss are potent psionics able to bend the forces of the universe to their will. Among their most powerful psychics are the templars, who can unleash devastating psionic storms, disrupt enemy systems, and even fuse together to form the nigh-unstoppable archons.
Note that the protoss we see aren't even in the zenith of their civilisation - they've been decaying since the zerg attacked Aiur, but still manage to kick ass.
TL;DR: it may seem like magic, but it is in fact a combination of very advanced technology and their innate psionic (telepathic/telekinetic/projection) abilities.
You might also want to watch this official video overview, which, again, attributes protoss' feats to advanced technology and "the power of the mind", the psionic abilities:
Regarding your last point, terran do use the same "magic" as protoss, just not to same extent. What you're looking for is called a "Ghost":
Born with incredible psionic potential, these individuals are recruited, quarantined, and put into training by the government from childhood. Ghosts learn to channel their psionic energies to augment their natural physical strength and endurance.
So psionic powers are not limited to protoss: terran and zerg can also use them, it's just that the protoss rely on them more than anyone else.
Nova (above), Tosh (before he went nuts), and Kerrigan were all Ghosts - their psionic abilities were to some extent innate (but not as powerful as the protoss, I think), but were later greatly enhanced by training.
It's not magic, it's "psionic"... which is essentially a scifi-flavored synonym for Magic. But note that the Protoss are not the only species with psionic abilities in the StarCraft universe. The Zerg hive mind communicates psionically and the Terran Ghosts also have psionic abilities. The Queen of Blades (Human/Zerg hybrid) possesses psionic abilities which are similar to those of the Protoss (in StarCraft 1 she has the Psi Storm ability which is otherwise only used by Protoss Templars). The Protoss just seem best at it.
Supernatural abilities aren't actually that rare in science fiction, especially in the more "soft science" universes. You have them in Star Wars ("the force"), Star Trek (various telepathic species), Babylon 5 (Telepaths which on a very high level can have telekinetic abilities, scientifically provable existence of "souls"), Warhammer 40k ("psykers") and countless others. Some people insist that only universes which use only plausible technology based on science should be called "Science Fiction" and any which also have supernatural elements should be called "Science Fantasy" instead. But the term never really caught on, because it would cover too many universes generally considered typical Science Fiction.