You're leaving out an important piece of info in the article
"She told the judge that it is for the vets' practice to prove that they did not discriminate against Ms Bailey's protected gender-critical characteristic, rather than for her to prove that they did.
'The claimant has shifted the burden of proof and... the defendant cannot satisfy the court that it did not do the act alleged or that the act was in no sense whatsoever because of the claimant's gender-critical belief,' she said."
Initially, Allison Bailey must present a prima facie case that she's been discriminated against. Once she does that, the burden shifts to the defendants (the vet's office) to prove that they did not discriminate.
Here is a good explanation
When you make a discrimination claim, you need to show the court evidence that you’ve been treated unfairly and that the reason you’ve been treated unfairly is because of a protected characteristic.....
When you make a discrimination claim, you need to provide the court with evidence from which it could decide that the discrimination took place. The obligation on you to provide this evidence is called the burden of proof.\
To satisfy your burden of proof, you need to show the court facts from which it could decide without any other explanation from the person or organisation you're taking action against, that you've been discriminated against...
If the court thinks you’ve shown enough facts to meet the basic legal test, it will conclude you’ve been discriminated against unless the defendant can provide a good enough explanation for your treatment. The burden of proof is said to shift to the defendant. They would have to show that your treatment had nothing to do with a protected characteristic.
https://www.citizensadvice.org.uk/cymraeg/Y-Gyfraith-a-Llysoedd/discrimination/taking-action-about-discrimination/what-do-you-need-to-show-the-court-in-a-discrimination-claim/