When you apply for pet insurance, at some point in the process you'll be asked to confirm that the information you've given is true and complete to the best of your knowledge. That confirmation is the declaration and by making it, you're taking on a legal responsibility for the accuracy of what you've said.
The declaration is what turns your application into a binding agreement. By declaring that your information is accurate, you're giving the insurer the basis on which to offer you cover. If that information turns out to be wrong, whether because you made a genuine mistake or because you left something out, the insurer can argue that the policy was based on inaccurate information.
This is not an invitation to paranoia. Most pet owners fill in their applications accurately and never have any issues. But it does mean that questions about your pet's health history should be answered carefully and honestly rather than skimmed through. If you're not sure whether something counts as a pre-existing condition, err on the side of declaring it and let the insurer make the decision.
The declaration also usually covers your agreement to the terms and conditions of the policy. Reading at least the key terms before you tick the box is worth doing rather than treating it as a formality.