The Stanleys were an important noble family in England, on whose support Richard III (of York) could not be sure.

Thomas Stanley, the head of the family, had married Margaret Beaufort, the mother (by a previous marriage) of the last Lancaster claimant to the throne, Henry Tudor.

Margaret had already in past conspired to advance the cause of his son; she was punished by the King, and put in custody ... under his husband Lord Stanley.

When Henry Tudor arrived in England in 1485 to claim the crown, Richard was unsure if he could count on Lord Stanley support or not. He did ask him to assemble his forces, and kept one of his sons Lord Strange, as an hostage.

During most of the battle of Bosworth the Stanley forces did keep apart, enraging King Richard. Worse, when the King personally led a charge against the position were Henry Tudor was observing the battle, the Stanley finally entered the battle, striking Richard column from the side.

Richard was killed almost within striking distance of Henry Tudor. One of the source that we have wrote that it was Lord Stanley that found Richard crown on the field of battle and gave it to Henry.

Source: I have read many books on the War of the Roses, one of the more recent ones is "The Road to Bosworth Field", by Trevor Royle.