Domestic violence is typically prosecuted as an assault charge (or sometimes other charges like uttering threats) and is proven like any other criminal case, with admissible evidence and proof beyond a reasonable doubt.
Many people don’t realize that testimony in court is evidence. All that is required to prove an assault charge is testimony from the witnesses that is believed by the judge. Corroborating evidence like photographs of injuries, medical records etc is not necessary to prove an assault charge. Sometimes text records, facebook posts and other digital evidence are also important.