Why were Ruiz and Montez arrested for slave trading?

Because they were knowingly buying and selling slaves that were obtained through the Atlantic slave trade which was outlawed at that point.

Yes, bringing slaves from Africa to Cuba was against Spanish law, and was a major plot point in the movie when Roger S. Baldwin proved it in court. I believe that Ruiz and Montez had already been arrested for something else in real life, and anyway, in real life Roger Sherman Baldwin was one of the top trial lawyers in the country, not the hungry ambulance chaser Spielberg made him out to be. The film implies he lost his business over this case, but think about it. If you want to hire a lawyer, do you care that he got some guilty guy off? No. You want the guy who took on a seemingly impossible case and won. This case made R.S. Baldwin governor of Connecticut and eventually United States Senator. His legal arguments won the case before the SCOTUS, not the Fourth of July oration by an ex-POTUS.

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I know it is an old post, but I realized nobody actually answered your question, stirling.

'Was bringing slaves to Cuba from Africa against Spanish law? Slavery was legal in Cuba but was the slave trade illegal': NO, by 1811, slave trade was forbidden by the British government and a treaty was signed between England and Spain, agreeing to it. So, yes, slavery was legal in Cuba, but slave trades became illegal for Spain and all of it's colonies (Cuba was a colony of Spain, therefore, bond to the same treaty by international rules). Besides, it was a Spanish ship doing the trade.

'Isn't that a matter for Spanish authorities then?'. No, because, international treaties are made between different countries and must be respected by them. When Spain violated it (by trading slaves), they became subject to international judgement, i.e.: from USA or England. USA Court judged them for violating the treaty between Spain and England.