Shakespeare is not the only playwright praised by people. However, he is one of the most famous and sometimes fame is a self-fulfilling prophecy. People have heard about the plays, so they read them to see what all the fuss was about. More than that, I think that Shakespeare's plays are great fun and we enjoy them for their valuable contribution to our literature.
With Shakespeare there is, indeed, a timelessness. Also, his stories are replete with intrigue, supersition, passion, daring, bawdiness, and everything else that makes for a recipe of universal appeal. Renowned Shakespearean critic Harold Bloom contend that William Shakespeare is Freud's teacher. If this be true, there is much of man that the Bard understood. Hence, his tremendous appeal.
Many English dramatists are praised, Marlowe and Jonson are two others who are praised, but it is true that Shakespeare gets more praise than others. This is because Shakespeare succeeded in writing plays of immortal interest--interest that continues from one generation to the next--of universal themes that concern everyone in every era; and of innovative and exquisite language. British school children have traditionally been surprised when reading Shakespeare to discover that his work is the origin of catch phrases heard all through life, like "the play is the thing ...."
He is certainly the most influential dramatist in English literature but certainly not the most praised. Christopher Marlowe is just as respected. So is Ben Johnson, and John Webster. On the other side of the pond there is quite a lot of respect for the works of George Bernard Shaw (although he is Irish), Oscar Wilde (another Irishman), and a myriad more.
However, a similar situation happens in Spanish literature. Miguel de Cervantes and Don Quixote de la Mancha, to the naked eye, may seem to enjoy nearly every attention given by academia. However, we also have Pedro Calderon de la Barca, Federico Garcia Lorca, and a myriad of luminaries who have placed Spanish literature on the pedestal that it so deserves.
I think that it is completely wrong to say that people only praise Shakespeare. He is certainly not the only playwright whose works are seen as classics. As to why he is praised so highly, I think that it is due to the power of the stories he chose to tell and the beauty of the language that he used to tell those stories. Both the plots of the plays and the actual words he put in the characters' mouths are important to his fame.
Shakespeare was one of the first masters of modern English. His brilliance was exceptional for the time period and continues largely unequalled today.
Shakespeare wrote mostly in iambic pentameter, a feat in itself. He chose words with specific sllyabication and rhyming pattern in order to follows the requirements of iambic pentameter. When he opted to deviate from iambic pentameter, it was for a specific literary purpose.
In addition, he wrote with layers of meaning. His unschooled audiences might understand the baldy meaning of his lines. Often, however, he wrote passages so they had multiple meanings, appealing to his diverse audience, from royal to poor working class.