The same way that you "prove" anything in the legal system: Present your case in front of a jury, while your opponent has the chance to rebut your claims. Keep in mind that the legal standard for civil claims is "more likely than not," as opposed to the more famous "beyond a shadow of a doubt" standard that is used for criminal cases.

So, you'll go up at trail and present a bunch of data points, and build a narrative around those data points. Look, you say, I had good reviews from my managers and glowing recommendations from my co-workers. Despite this, I was fired shortly after I Did A Thing. Is this a coincidence or retaliatory? Your employer would attempt to provide evidence that your work quality actually dropped off, you would attempt to present evidence that either the quality stayed the same, or else that goalposts were moved for no good reason.

Note that both parties have the ability to compel various types of evidence to be produced. You would be able to subpeona the employer for emails between bosses, or manager notes, or reviews and such. You could be able to compel people to testify around a variety of facts.

... if this process sounds rather onerous, now you know why trials take a long time and many parties prefer to settle privately rather than engage in the full court process.