DVK makes a good point that they were probably afraid of Mrs. Weasley's wrath if they used magic, but there may be more basic reasons why they didn't use magic. In Chapter Seventeen of Half-Blood Prince Harry thought that the Ministry could actually detect underage magic, until Dumbledore disabused him of this notion:
"But how come the Ministry didn't realize that Voldemort had done all that to Morfin?" Harry asked angrily. "He was underage at the time, wasn't he? I thought they could detect underage magic!"
"You are quite right — they can detect magic, but not the perpetrator: You will remember that you were blamed by the Ministry for the Hover Charm that was, in fact, cast by —"
"Dobby," growled Harry; this injustice still rankled. "So if you're underage and you do magic inside an adult witch or wizard's house, the Ministry won't know?"
"They will certainly be unable to tell who performed the magic," said Dumbledore, smiling slightly at the look of great indignation on Harry's face. "They rely on witch and wizard parents to enforce their offspring's obedience while within their walls."
The scene you ask about occurred in Chapter Sixteen of Half-Blood Prince; thus, at that point Harry (and presumably Ron as well) was still under the impression that the Ministry would know if they did magic.
Another more basic reason for not doing it via magic is that they probably lacked the ability to do so. Harry's magical talent seems to be almost entirely in dueling spells, and Ron's magical talent is... closer to non-existent. Magically peeling sprouts is presumably something that neither of them have done before, and it would be unlikely that they would suddenly figure out how to do it and do a good job of it. Consider that several months later (Chapter Twenty-Four) we have the following instance of Harry and Ron utterly failing at a new charm:
"Flitwick," said Ron in a warning tone. The tiny little Charms master was bobbing his way toward them, and Hermione was the only one who had managed to turn vinegar into wine; her glass flask was full of deep crimson liquid, whereas the contents of Harry's and Ron's were still murky brown.
"Now, now, boys," squeaked Professor Flitwick reproachfully. "A little less talk, a little more action... Let me see you try...."
Together they raised their wands, concentrating with all their might, and pointed them at their flasks. Harry's vinegar turned to ice; Ron's flask exploded.
"Yes... for homework," said Professor Flitwick, reemerging from under the table and pulling shards of glass out of the top of his hat, "practice."
Or consider that when Harry was about to come of age (Deathly Hallows Chapter Two), and therefore be able to do magic all the time, he realized that he didn't even know how to do a simple healing spell:
It was stupid, pointless, irritating beyond belief that he still had four days left of being unable to perform magic…but he had to admit to himself that this jagged cut in his finger would have defeated him. He had never learned how to repair wounds, and now he came to think of it – particularly in light of his immediate plans – this seemed a serious flaw in his magical education. Making a mental note to ask Hermione how it was done, he used a large wad of toilet paper to mop up as much of the tea as he could before returning to his bedroom and slamming the door behind him.
So in short, there may be a few reasons why they didn't use magic in the Burrow. They probably wouldn't have been able to do it anyway, they didn't yet know that the Ministry couldn't detect their magic, and they would have been afraid of Mrs. Weasley anyway.