Of British-English origin, the phrase to cross the floor is used of a member of parliament, and means to join the party opposed to one’s present party.
—Cf. also hung parliament and the Cat-and-Mouse Act.

In the House of Commons, government and opposition parties sit opposite and facing each other, and the phrase refers to the practice whereby a member of parliament literally walks across the floor to join another party on the other side.

House of Commons Chamber—photograph UK Parliament:

 

These are the earliest occurrences of the phrase to cross the floor that I have found, in chronological order:

1-: From the transcript of the speech that Thomas Creevey (1768-1838) delivered in the House of Commons on Thursday 14^th March 1822—as published in Hansard’s Parliamentary debates:

By act of parliament, the country is to have a share of the company’s profits. I would say to the company, then—still treating it as a matter of economy—“the less members you pay for, the more money the country will have to receive of you.” In regard to these commissioners, the real question is, not as to who pays them, but as to their votes. Look, for instance, at the learned doctor, now on the other side (Dr. Phillimore ¹); who cares by whom the salary of the learned doctor is paid, or what its amount is? The company pays him 1,500l. a year; but, by the single circumstance of his having crossed the floor, with his friends, the country has still to pay 1,500,000l. a year, the amount of the salt tax. But a very little while ago, the doctor was the champion for the repeal of the salt tax; he was the foremost in the field against it; he was the leader; nothing could stop him; he must and would repeal it forthwith. [Cries of “Order, order,” accompanied by laughter.] If I am wrong, and gentlemen will suggest any other name by which it is proper that I should call the doctor, I will avail myself of it; but I know of no other name but the doctor. The learned civilian then, I say was the champion for the repeal of the salt tax; but now, the company pays him 1,500l. a year, he gives them his valuable services cheap as dirt; but I am not going into the question whether those services have been purchased at a cheap or dear rate. Suffice it to say, that he and his friends have lost the country at one slap, 1,500,000l.

¹ Joseph Phillimore (1775-1855)

2-: From the Yorkshire Gazette (York, Yorkshire, England) of Saturday 10^th August 1822:

We were not aware, till we read Mr. R. Martin’s speech, that the great arithmetical luminary, Mr. Hume ², had once been an adherent of Ministers. We knew, indeed, that he had before sat in the House of Commons, but on which side he sat, had never occupied our thoughts. We only wonder that the Chancellor of the Exchequer could have let such a financier slip through his fingers, and we dare say by this time he bitterly repents his negligence. One thing, however, is worth remarking. Mr. Hume, as a Ministerial Member, was nobody; absolutely a cipher; a mere non-entity; but the moment he crossed the floor of the House of Commons, and joined the pigmies on the other side, he became, all at once, a very important personage. In a nation of blind people, says the proverb, a one eyed man would be King.

² Joseph Hume (1777-1855)

3-: From George Tierney’s ³ address to William Plunket ⁴, during a debate in the House of Commons on Friday 11^th February 1825—as published in Hansard’s Parliamentary debates:

In the autumn of that very year, the right hon. and learned gentleman took office. He (Mr. T.) was not intending to impute bad motives to him, by any means, for this acceptance of office. He was ready to believe, on the contrary, that the right hon. and learned gentleman accepted office with the most honourable and legitimate intentions. But he took it; and at the same time strode over from that side of the House (the opposition side) to the other. Why the right hon. and learned gentleman was made attorney-general for Ireland, except for his conduct with respect to the Roman Catholic claims, he was at a loss to know. But, if it was on that account that he was chosen attorney-general, it might be supposed that if the right hon. gentleman persisted in his support of the same question, the government would make him lord chancellor. He should say to that right hon. and learned gentleman—“It is most extraordinary that you who kept to your opinion in adversity—when you sat on this side of the House, I mean—[a laugh], should not keep to it in prosperity, when you might adhere to it with effect. If while you sat among us, you were consistent in this respect, why should you be otherwise the moment you crossed the floor to the other side?”

³ George Tierney (1761-1830)
⁴ William Plunket (1764-1854) was then Attorney-General for Ireland.

In some countries, such as Australia, the phrase to cross the floor is also used to mean to vote against one’s own party—as illustrated by the following from The Age (Melbourne, Victoria) of Wednesday 30^th May 1979:

Three Libs cross floor

Canberra.—Three Liberal backbenchers crossed the floor and voted with the Opposition during debate last night on legislation regarding listening devices.
Sir William McMahon and Mr. Hodgman (Tas.) crossed the floor twice, and Mr. Wilson (SA) once.
Mr. Don. Cameron (Lib., Qld.) twice abstained from voting on the same legislation, the Customs Amendment Bill, and Mr. Chapman (Lib.. SA) once.
Mr. James (Lab., NSW) also crossed the floor and voted with the Government.
[…]
Three Government backbenchers got up to speak on an amendment moved by Sir William, but the Deputy Government Whip, Mr. Hodges, moved to gag debate. Sir William and Mr. Hodgman crossed the floor in protest and Mr. Cameron abstained.
The gag was carried and Sir William’s amendment was lost.
Then Mr. Wilson moved an amendment which said that information on a crime other than a narcotics offence should not be communicated to other agencies if the information was obtained through the use of bugs.
Mr. Wilson voted with the Opposition. Mr. Hodgman and Sir William crossed the floor for the second time.
Mr. Cameron abstained again and was joined by Mr. Chapman.
Mr. James crossed the floor and voted with the Government against the amendment, which was lost.

In extended use, the phrase to cross the floor means to change sides on an issue, to reverse one’s opinion or position—as in the following, by Emma Rowley, published in The Daily Telegraph (London, England) of Wednesday 26^th January 2011:

Bank of England MPC minutes: Martin Weale joins Andrew Sentance to vote for interest rate rise

A senior Bank of England policymaker has crossed the floor to call for interest rates to rise, minutes for the Monetary Policy Committee’s latest meeting have revealed.
Martin Weale, an external member of the nine-strong committee, unexpectedly joined Andrew Sentance, his most ‘hawkish’ colleague, to call for the base rate to go up from its record low of 0.5pc to 0.75pc [&c.].