Iago's a slime ball. It must be something in human nature that we all love a good villain. He is not exactly a sympathetic character, but by the end of the play we might not sympathize much with Othello either.
My students always love Iago... although many love to hate him rather than actually loving him.
I think part of the charm is that Iago is just so familiar. We'd all met someone who feels slighted, who feels left out, who feels like people are talking behind his back. Many times, we have been that person. But Iago does what most people leave in the realm of fantasy... he gets revenge. In many ways, I think Cassio almost deserves it. He is clearly out of his league and his bookish education is not really up to the task of the army or he would not be so easily led and so easily manipulated. Iago takes him down in record time. However, the tragedy is that Iago then turns on our hero/heroine who are truly not deserving of his hatred.
So there's this weird dichotomy with part of the audience feeling Iago's frustration and gleeful joy at proving to Othello that Cassio was a poor choice and part cringing at how far Iago then takes his revenge. He's a great character to read, though!
Iago confides in the audience. He makes the audience an ear for his plotting, his scheming, and his meddling. He tells us what we are about to witness, and we cannot help but be baffled by how successful he is in carrying out his debauchery, following through with his intent. We know what will happen and yet there is nothing we can do. All those who have grown sympathetic to Othello and his beloved Desdemona, cannot help but be compelled by the sense of helplessness Iago creates for the audience. He is skillful, cunning, duplicitous. He is an artist of evil as Shakespeare is of the soliloquy.
I'll go a little "darker" and say that Iago intrigues readers because there is evil and bad in ALL of us and that side appeals to us in a very screwed-up way. Iago does things that perhaps some of us have wanted to do...interfere in a relationship of which we are jealous, get back at someone who is more successful than we are, etc. For those of us who do NOT go to the "dark side," we can at least live it vicariously through Iago!
As despicable as Iago is, Shakespeare has created a character that intrigues the audience. Other Shakespearean villains fall victim to ambition, choosing to do evil deeds in order to promote themselves. Iago has no such plans--he, interestingly enough, seems to enjoy being the bad guy, seeing "sport and profit" (1.3.287) in his manipulation of Othello. For someone who practices deception, he is surprisingly transparent--he loves being the villain.
On some level I think we almost admire Iago. He is, after all, quite good at what he does. With a few well placed words and images, he exploits Othello's weaknesses and faults. There is great power here--terrible, no doubt--but we are drawn to Iago because of it.
Iago is an incredibly charismatic character, he is charming, witty and highly intelligent. He has a way with words and this manipulation of words not only works on the characters within the play but also on the audience.
Human nature sees us drawn to many situations and people who are abhorrent and Iago is no exception. That's why Shakespeare's works are so timeless; because he is an amazing observer of human nature.
Iago is a genius of psychology and manipulation. With the minimum of effort he sculpts peoples' lives as he wishes. Perhaps in reality he wouldn't be able to 'make his puppets dance' so perfectly, but it's fascinating watching him. And we reluctantly like him because while he shows us that he's surrounded by gullible idiots, he treats us (the audience) as intellectual equals. He takes us behind his 'good honest Iago' mask and says, "watch me get that smug Cassio drunk and punch Roderigo. Then I'll fix it so he gets a big slap off Othello, but I'll look like the decent bloke who tried to calm everything down. This should be fun, here we go!"
And to start with we feel it is fun (sort of). Wrong? Yes. Fun? Definitely. Roderigo is a witless snipe. Othello gave a job to Cassio which Iago fully deserved and Cassio did not. Othello may have slept with Iago's wife. His victim's are not Angels. And Iago is very charming and likeable when he wishes.
But then it all starts getting a bit heavy. If he'd stopped after the drunken fight and taken Cassio's job from him, we might forgive him. But he doesn't stop and we gradually discover that our new friend is a twisted psychopath and that we our way out of our depth. Suddenly we find ourselves accomplices to terrible crimes. What started as fun ends as horrible hellish cruelty and murder.
You're never bored when Iago's around!