Why did they arrest Anne and then behead her for adultery?

Because it was easier than divorce.

In the times of King Henry VIII (And actually till quite recently), it was a common conception that if a woman couldn't produce a (preferably) male offspring, it was her fault. So, the logic for King is simple - if Anne didn't give birth to a boy, she was somehow faulty, the miscarriage confirmed his suspicion. So what was the logical solution? Finding another mother.

However, the king of England cannot just produce the future ruler with a maid - the child must be from a legitimate relationship with his wife AND there cannot be any doubt who is the father. So, in short - he has to get rid of his wife.

Now, King Henry is a Catholic and in Catholicism, there is no such thing as a divorce, since the marriage is considered sacred. Marriage can be annulled (made invalid) if (since we are talking about King here!) the Pope himself will agree to do so. In the case of his first wife, Catherine of Aragon, he tried to convince the Holy See that this marriage was invalid, for example by citing Levictus 20:21 ("If a man marries his brother's wife, it is an act of impurity; he has dishonoured his brother. They will be childless."). This was a long, thankless and EXPENSIVE work...

In the case of Anne, he couldn't count on another annulment and with young Jane Seymour on the horizon, he had to act quickly. So he charged his (current) wife with "treasonable adultery" and executed her (since banished/imprisoned Anne would still be his wife), making way for a new one.

It is worth pointing out that the issues with getting divorced by King Henry VIII were one of the main reasons for the English Reformation, which put the King of England at the head of the Church.

Update: As @tbrookside pointed out, Henry already was the head of the Church of England during the marriage to Anne, so he could dissolute his own marriage. However, by doing so he would alienate the still big Catholic part of the country. Since his first wife had died by the time of Anne's fall from grace, by killing her he would achieve a "clean slate" and his next marriage would be legitimate even in the eyes of Catholics, reducing the discontent in the country and increasing his support. So he had accepted Cromwell's proofs of Anne's treason.