Why aren't underage wizards allowed to do magic outside Hogwarts?

Question

Why can't they use their holiday to secretly practice the spells they've learned?

Answer 1

Considering how it's enforced, it's likely it's to keep magic secret.

The rule is (in effect) for Muggle-borns or wizard children who will be around Muggles - it's unenforceable near adult wizards. In this case it makes sense. Their parents would of course have been made aware of the wizarding world, but the Ministry wouldn't want some Muggle-born kid deciding to do magic to impress the Muggles, or thinking it's okay to do magic near them. Children are terrible at keeping secrets, so it's probably easier to ban them from doing magic and monitor them so they could handle any breaches of the Statute of Secrecy that wizard children around Muggles could cause.

The rule against underage magic is only really enforced when no adult wizards are around.

It's technically still against the rules for an underage wizard from a wizarding family to do magic outside of Hogwarts, but anywhere there are adult wizards around, it becomes unenforceable since there's no way of knowing who did the magic, and if it was the child or a qualified wizard.

“He was under age at the time, wasn’t he? I thought they could detect under-age 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 under age 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.” _{^{- Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)}}

This means that in wizarding households, as well as wizarding locations like Diagon Alley or Hogsmeade, the ban on underage magic would be impossible for the Ministry to enforce, so they trust the parents to control their children. If the parents let their children do magic, however, the Ministry won't know and has no way to punish them.

Answer 2

Because they are not fully qualified, so it's seen as the best solution to prevent them from doing things wrong. Think of it a bit like a drivers license education: you're not allowed to practice on the streets on your own, because you're not yet qualified to drive a car in public. Doing it under the watchfull eyes of an instructor is another thing ...

Answer 3

Because of the Decree for the Reasonable Restriction of Underage Sorcery

Paragraph C of the Decree for the Reasonable Restriction of Underage Sorcery prohibits underage wizards from doing magic outside of school.

We have received intelligence that a Hover Charm was used at your place of residence this evening at twelve minutes past nine.

As you know, underage wizards are not permitted to perform spells outside school, and further spellwork on your part may lead to expulsion from said school (Decree for the Reasonable Restriction of Underage Sorcery, 1875, Paragraph C).

_{(Harry Potter and the Chamber of Secrets - Chapter 2)}