What does it mean for a voting law to be "racist"? For the purposes of this thread, at least when it applies to a United States law, "racist" should be considered short-hand for violating a provision of the United States Constitution or the United States Voting Rights Act.
In the US a judge (or panel of judges) is the person under US law that is tasked with determining whether or not a law violates the Constitution or the Voting Rights Act. This fact should not be confusing or controversial.
As hopefully everyone posting in this thread realizes, the US has a sordid history of discriminatory laws designed to disenfranchise blacks and other citizens. Right after the civil war, a series of amendments to the Constitution were passed that gave blacks the right to vote. Sadly, soon after that eleven southern states passed laws designed to make it difficult for blacks to actually vote in an election. These took the form of poll taxes which required prospective voters to pay a fee to register to vote.
In due course these poll tax laws were stricken down by the US Supreme Court as violating the equal protection clause of the 14th amendment of the Constitution. Essentially the justices ruled that how wealthy a citizen was should have no bearing on his/her right to vote.
Most modern voting court cases are decided upon via the Voting Rights Act passed in 1965 extending the protections of the 14th and 15th amendments of the Constitution. Section 2 of the Voting Rights Act explicitly prohibits voting practices or procedures that discriminate on the basis of race, color, or membership in one of the language minority groups identified in Section 4(f)(2) of the Act.
In 1982, Congress amended Section 2 to provide that a plaintiff could establish a violation of the section if the evidence established that, in the context of the "totality of the circumstance of the local electoral process," the standard, practice, or procedure being challenged had the result of denying a racial or language minority an equal opportunity to participate in the political process.
Thus, the Voting Rights Act prohibits not only laws (practices, procedures, or standards) that were designed to be discriminatory (prohibited by the 14th and 15th amendments), but also laws that are in fact discriminatory by their results (the now famous "results test") whether or not the law was originally designed to be discriminatory.
In every case of this type that has been decided upon there are hundreds if not thousands of pages in the trial history (briefs, etc.), many of which purport to show the disproportionate effects of the law in question. The Texas case has innumerable documents available to the public should anyone here be interested in reading them. Here is a link to that court case's history, documents, briefs, etc.
https://www.brennancenter.org/legal-work/naacp-v-steen