Why did Shakespeare write sonnet poetry?

Though it can be hard to imagine in our modern age, in which all of the most respected writers are novelists or authors of other long form fiction, in Shakespeare's day, the poem was considered the height of writing skill and the universally regarded standard by which a writer was judged. To put it simply, if a writer in the time wanted to be taken seriously, they needed to put considerable effort into verse writing. To the question of why Shakespeare set out to write sonnets from an artistic point of view, it seems that his entire collection is a multi-faceted and comprehensive exploration of the concept of love, one that considers every perspective possible. Shakespeare was nothing if not a romantic, and seemed to want to convey every expression of love in his verse, even the base and misguided sort. However, just like his plays, there is no evidence whatsoever as to what Shakespeare's plans for publication were, if any. Any answer to this question is ultimately a matter of conjecture.

It is thought that Shakespeare composed and revised the sonnets between the late 1590s and 1605. Not much is known about Shakespeare's involvement in their publication, which means that his reasons for writing them will always be a matter of speculation.
However, by examining the themes of the sonnets and studying the form that the bard helped to popularize, informed guesswork can provide some satisfying conjecture. Because his plays were written using iambic pentameter, it is fair to say that Shakespeare enjoyed the challenges of working with verse. He would have been familiar with the sonnet form through reading Henry Howard's translations of Italian sonnets, and with his creative mind, it seems natural that he would alter the form to suit his thematic needs. There is plenty of evidence in the plays, for example, that demonstrates Shakespeare's wit, such as the wordplay that marks the characterization of Hamlet. In Shakespeare's Sonnet 130, the speaker spends the first twelve lines describing the many shortcomings of his beloved, only to then profess his love for her in the final couplet.
Shakespeare longed to be accepted as a serious literary figure, and while his plays did entertain a broad spectrum of the audience of his day, the poets were the time's revered writers. His meditations on life, love, and the passage of time are timeless and universal and the sonnets allowed him to express briefly and eloquently his insights into those themes.

Shakespeare wrote sonnets because they were a respected poetic form in his time period. A person who wanted to be taken seriously as a literary figure would write sonnets or other forms of poetry.
Shakespeare is primarily remembered as the world's greatest playwright, but his plays would have offered him less prestige while alive than his sonnets. Shakespeare took care with his sonnets in a way he did not with his plays. We do not know whether Shakespeare cooperated with the 1609 publication of his sonnets (he may or may not have), but we do know he never put out an authorized version of his plays.
Shakespeare spoke of love and time in his sonnets. He also created what is now called the English or Shakespearean sonnet form. His sonnets consisted of three four-line stanzas followed by a couplet (a verse of two lines). The older form of sonnet, the Petrarchan, consisted of one eight-line stanza followed by a six-line stanza.