Aunt Polly finds out that Tom had been out swimming when Sid alerts her to the thread on Tom's collar. Instead of the original white thread she used to sew on Tom's collar, the thread is now black. This means that Tom took off his collar before he went for a swim. After the swim, he sewed the collar back on, except that he used black thread, instead of white (the color of the thread Aunt Polly originally used).

According to the text, Aunt Polly suspects that Tom skipped school to go swimming. However, she can't come up with clear proof to confirm her suspicions. After all, the shirt collar appears to be securely sewn onto Tom's jacket. The only evidence that Aunt Polly can point to is that Tom's hair is wet.

As for Tom, he tells Aunt Polly that he merely wet his hair with water. Aunt Polly gives up her line of questioning. However, Sid (Tom's half-brother) pipes up with “Well, now, if I didn't think you sewed his collar with white thread, but it's black.”

At this, Tom quickly makes a getaway. Sid's revelation upsets Tom, and he consoles himself with the fact that he'll make Sid pay for outing him.

How did Aunt Polly discover Tom swam in "The Adventures of Tom Sawyer"?

Aunt Polly is suspicious of Tom because of his wet hair.  During dinner, she asks a series of "leading" questions trying to get him to confess to having skipped school to go swimming:

“Powerful warm, warn't it?”

“Yes'm.”

“Didn't you want to go in a-swimming, Tom?”

A bit of a scare shot through Tom—a touch of uncomfortable suspicion. He searched Aunt Polly's face, but it told him nothing. So he said:

“No'm—well, not very much.”

The old lady reached out her hand and felt Tom's shirt, and said:

“But you ain't too warm now, though.”

Tom keeps avoiding her questions.  He claims to have used the water pump on his hair.  The tricky part is his shirt collar.  Aunt Polly had sewn it.  If he got wet with the water pump, it would still be intact.  If he had removed his clothes to swim, it wouldn't be.

At first, Aunt Polly is reassured by the look of the sewn color.  But it is Sid, Tom's brother, who finally rats out Tom.  He points out that the thread is black, but Polly had used white thread.   Thus, she knows is has been re-sewn by Tom.

How did Aunt Polly try to find out if Tom went swimming?

Tom was supposed to go to school, but as it's such a lovely hot summer's day, he decides to play hooky and go swimming instead. Aunt Polly is suspicious and tries to get Tom to own up at dinner later that evening. But Tom adamantly insists that he didn't play hooky. If he'd gone swimming, he says, he would have had to remove his shirt collar, undoing Aunt Polly's sewing. But look—the collar's seam is unbroken, and so Tom couldn't have gone swimming.

Aunt Polly apologies to Tom for her suspicions. But trust Sid the snitch to mess things up for his cousin. He notices that Tom's shirt collar is being held together by white thread, even though Aunt Polly used black thread when she sewed it that morning. Tom's plan has been rumbled; he used white thread to sew his collar back on after finishing his swim, thinking he'd be able to fool Aunt Polly. But thanks to Sid's eagle eye, Tom is caught in a lie.

How does Aunt Polly discover Tom went swimming?

This happens in the first chapter of the book.  Tom goes swimming instead of going to school and he thinks he's going to get away with it.  But then his cousin Sid tells on him and that's how Aunt Polly finds out.

Aunt Polly had sewn Tom's collar shut that morning.  Tom figured out that he could just undo the stitches, go swimming, and then stitch it up again.  And that's what he did.  But then Sid pointed out that the thread is now black whereas Aunt Polly used white in the morning.  Once that is pointed out, Aunt Polly knows what Tom did.