"Liability coverage can even pay for damages caused accidentally by you or your family members. For example, if your child accidentally throws a baseball through your neighbor's living room window, your liability insurance plan could cover the damages." ALLSTATE RENTERS INSURANCE
That came up as the first hit on google with the search: what does renters insurance cover. Here's some others
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Property damage to others
Rental insurance coverage for your apartment or house follows you wherever you go, so if you accidentally break or damage someone else's property, your policy can help pay to replace it. ESURANCE RENTERS INSURANCE
Renters insurance also provides coverage for personal liability relating to injury sustained by any non-resident at your address, including payments for medical expenses – as well as litigation which may be brought against you. Liability insurance will also protect you in the event anyone brings a lawsuit against you for damage to another person’s property.INSWEB
Besides providing protection for your personal property, renters insurance also provides liability protection. That coverage extends to injuries and other damages visitors sustain while on your property as well as damage you may inadvertently cause to the building. For example, if a guest in your home falls and gets injured, a renters insurance policy would cover the costs of the medical bills as well as legal representation if you get sued. MAINSTREET.COM
These quotes are all directly from the top several hits on a quick google search. It seems unanimous. Renters insurance pays if the renter damages the rental unit.