Firstly, always contact your insurer as soon as possible after noticing the damage or theft. As per the question above, when claiming, you may have to pay towards repairs and replacements, known as an 'excess', so check your policy for the full information.
You'll need to provide details of the circumstances surrounding any loss or damage, plus any evidence of that. Take photographs of the damage to your home, contents or car, or film the footage. This may help provide proof.
If your possessions have been badly damaged or stolen, any photographs of you with a particular item when undamaged, or held by friends or relatives, will demonstrate you owned it. You can also use receipts, credit card bills or bank statements as evidence.
We have step-by-step help in our Making a claim guide.