We first meet Will as he hangs out in his tiny apartment. When he's not reading, he's hanging out with his friends (like Chuckie, who comes to his door to get him) or working as a janitor mopping the hallways of MIT.
Nothing too special, right?
Meanwhile, a professor at MIT named Gerald Lambeau puts an extremely complicated math problem on the hallway chalkboard to see if any of his students are up to the challenge. Turns out, they're not. But, of course, Will knows how to solve it, and he writes out the answer on the chalkboard when no one's watching. You know, just for complicated-math-problem kicks.
After finding Will's answer, Professor Lambeau takes things up a notch. He leaves another problem on the board, one that only a few of the world's greatest minds can solve. And... that's when he catches Will. Yep: The janitor is the one solving the math problems. Lambeau tries to go after him to find out who he is.
Meanwhile, Will gets himself into some trouble with the law when he assaults a police officer. Combined with his mile-long rap sheet, this assault puts Will on a one-way street to lock-up. But Lambeau tracks him down at the last second and makes a deal with the judge: Will can stay out of prison as long as he works with Lambeau on math every week—and gets some therapy on the side.
Will agrees to therapy, but mentally destroys the first five therapists Lambeau pairs him with. (Montage, anyone?) Finally, Lambeau reaches out to an old buddy named Sean Maguire, who's from the Southie neighborhood of Boston… just like Will.
At first, Will seems like he's too much for Maguire. Within minutes of meeting each other, Will realizes that Maguire has a lot of sadness around his memories of his wife, so Will pokes and prods at it until the therapist grabs him by the throat and threatens him.
Eek. What would Freud say?
After Maguire's had a chance to regroup, he takes Will to a park bench and tells him that it doesn't matter how smart he is or how much he's learned from books. All that matters is that Will is too chicken to live life on any terms but his own, which means he'll never know true love or experience the best things in life firsthand.
And that's how you make a speech. (Kind of a famous scene, BTW. No wonder why.)
When he's not in therapy or in his math sessions with Lambeau, Will hangs out with a girl from Harvard named Skylar. Things go well at first, but Will is an orphan and has a deep-seated fear that, eventually, anyone he loves will walk away from him. So he makes a preemptive strike and pushes Skylar away before she can get a chance to hurt him.
Tensions rise between Professor Lambeau and Will's therapist, Sean Maguire. See, Maguire thinks that Will should figure out for himself what to do with his life, but Lambeau won't be satisfied with any result that doesn't have Will using his gifts as a mathematician. Eventually, the two get into a huge blowup fight and Lambeau storms away, calling Maguire a loser and a failure.
Meanwhile, Will keeps working at a construction job that he got through his buddy, Chuckie. One day, Chuckie confronts Will about how he's wasting his intelligence hanging around construction sites. But Will says it doesn't matter if he's smart—as long as he's happy. Chuckie insists that Will owes a debt to all the other men working on the construction site who wish they had the gifts Will does.
Chuckie's wish? That he'll show up at Will's house one day and find Will gone.
In an effort to help Will get over his abandonment issues, Sean Maguire tells him the story of how he passed up one of the greatest baseball games ever to go on a date with the woman who would one day be his wife. He tells Will that he never regrets being with his wife, despite all of the pain he still feels after losing her to cancer. After he sees photos of the injuries Will got as an abused child, Maguire walks up to him and repeats the phrase, "It's not your fault" until Will breaks down and weeps.
Turns out our boy's a little vulnerable.
In the final moments of the movie, Will takes a job with one of the think tanks that Professor Lambeau set him up with. But when Chuckie shows up at his door... he's not there. He left Boston and is headed for California in his car, hoping to get Skylar back (she's at Stanford). Before going, he leaves a note in Sean Maguire's mailbox saying that he had to "see about a girl," which is the same line Maguire used on his friends when he first met his wife.
Pass the tissues.