Good question! A couple of points that may be useful and / or interesting.
First, he didn't actually write in rigid iambic pentameter. The sonnets, mostly yes, but that is a more rigid form. Though even there, he sometimes strayed. One of them has 15 lines, for instance. We don't have any of the plays in Shakespeare's own hand, but the earliest published versions, the First Folio, often show not only unusual versions of some speeches--compare the First Folio "Romeo, Romeo, wherefore art thou Romeo" speech with most other editions, for instance--but also surprising line breaks. Based on those, he didn't always finish lines with five feet, even if we take into account lines that are finished by other characters. Modern editors have taken it upon themselves to "correct" the text. I don't mean to say they're just making stuff up--there's often good reason to suspect the authenticity of the Folios, since they were published 7 years after Shakespeare died, and were often based on prompt-scripts or actors' memories of the lines--but it is possible that the editors created a consistency of meter that Shakespeare himself didn't intend.
Second, individual lines or feet also clearly stray, with or without editorial involvement. King Lear's "Never never never never never," is, for instance, a perfect trochaic pentameter line. For that matter, "To be or not to be," starts out with three perfect iambs, but then there's a caesura depending on whether you read "that is" as a trochee or an iamb ("THAT is the question," or "that IS the question"), and then "the question" is an iambic foot with a so-called feminine ending, an extra unstressed syllable. So the most famous line in all of Shakespeare is actually an 11-syllable line. Macbeth's "You should be women, and yet your beards forbid me to interpret that you are so" is arguably even harder to scan. All of which is to say, the notion that "Shakespeare wrote in iambic pentameter" is a guideline, not an absolute rule that he followed without deviation.
Third, you see it as a restriction, but in general, iambic pentameter is actually pretty easy, at least in English. English tends to fall naturally into iambs (or trochees, which are just backwards iambs), and if you try scanning any piece of text in English, while it while hardly ever falls into perfect pentameter, I think you'll find more iambs than anything else. For instance, a line like, "I tried to call you but you never answered" scans the same as "Is this a dagger which I see before me?", but you certainly wouldn't call it forced syntax. This is as opposed to, say, Ancient Greek, which has far fewer consonant clusters than English, and I believe longer words overall, and which tended to be spoken more quickly, which is why most Ancient Greek plays were written in dactyllic hexameter (BUM-ba-da BUM-ba-da BUM-ba-da BUM-ba-da BUM-ba-da BUM-ba-da ), though again, without perfect rigidity.
Because of that, iambic pentameter may have helped him write faster. Remember we think of him now as a poet, but in his own time, he was a playwright. Oh, he dabbled in poetry, sure, but two long poems and a sonnet sequence would not have impressed "real" poets like John Donne or John Milton as an output. He wrote for the theater, a mostly disrespected, popular medium (maybe think sitcoms today). And he wrote them mostly on commission. Anyway, writing within a given form can be easier, and therefore faster, than trying to write verse without one. Amy Poehler can be as creative as she wants, but when the new episode of Parks and Rec was due, it was due, so I bet knowing that whatever she wrote had to fit into 21 minutes with four commercial breaks helped her focus her energies. Similarly, Shakespeare probably picked the form that would help him get the play done on schedule so he could buy a round of ale at the pub, since he couldn't really afford to wait for the muse to smile on him (which may explain Pericles).
And while it may not have been a requirement for plays, he was not alone in using this meter. Try this famous line from his contemporary, and probably bar-buddy, Christopher Marlowe: "Is this the face that launched a thousand ships?"