I'm going to use your tense (was, not is) for this answer and avoid why he continues to be such a big deal today.

Shakespeare wrote at a time of great literary changes, especially in terms of drama. After a 'golden age' of classical dramas in the Roman Empire, the entire art form decreased in output and prestige. During the medieval time period most of the extant European pieces were liturgical (like the Bendiktbeuern Christmas Play) or educational (such as the works of Hrosvitha).

In England there were really only two genres of plays until the mid 1500's, the morality plays (which were extended allegories dramatizing the struggle of moral characters, like Everyman or Mankind) and the cycle plays (which were performed yearly and charted almost the entire Bible, usually spending a whole lot of time on the Passion of the Christ). There is a late 1400's play called the Play of the Sacrament that is neither morality-focused nor a Biblical tale, but it is a fiercely antisemitic conversion narrative, not exactly what we view as modern drama.

So Shakespeare begins his career in the late 1580's, just a few decades after the dawn of secular drama in England. Playhouses start to become a really big deal, both as popular entertainment for the common man and for courtly audiences. Different acting troupes started to receive patronage from lofty figures (Blackfriars theater, an indoor well-to-do theater was home to an all-child troupe patronized by Queen Elizabeth herself). Publishing was becoming a huge deal, and with Ben Jonson's 1616 Works it became normative to fashion what was once a rather lowly profession into a status-symbol, the idea of an author-figure came more and more to the forefront.

Shakespeare himself was a shrewd businessman, a partial owner of his company, an accomplished actor, and an unqualified talent. His plays often catered to certain audiences at the time and he often chose culturally appropriate topics (such as his wildly popular two quadrilogies of history plays), but in reality he has just been perceived as a better writer than those around him. He also wrote more than other playwrights of the time (at least 39 surviving plays), attesting to a strong work ethic and drive (Jonson, for example, was criticized for writing just one play a year).

So he was very good at what he did and he did what he did at a time when what he did was a very hot new thing. The publishing of his works in the first folio of 1623 helped cement his prestige and then once the 18th century hit literary critics and tastemakers such as Alexander Pope and later on Samuel Taylor Coleridge adored his work, building his reputation even greater. As to exact reasons why he was selected as the pinnacle of English literature, I think that's impossible to say. I suppose I could speculate, but that's past the purview of this sub. Hopefully I've given some context on the unique literary situation of the time, though.