The Offshore Account is used for purchasing contracts on Crime.net and, only during limited-time events, offshore paydays. As of Update #22, accumulation and expenditure of large sums of Offshore money is part of the conditions required to ascend each level of Infamy (requiring the player to spend $200,000,000 per ascension).
At the end of a successful heist, 80% of the player's earnings are deposited into the offshore account, while the last 20% becomes "spendable" cash. Prior to Update #11, the split was 95/5.
The Offshore Account is securely laundered, and deposited into untraceable accounts in nations that exercise banking anonymity and discretion. The total amount is visible on the screen shown in the Safehouse Vault. Meanwhile your Spending Cash is stacked in the form of physical US Federal Reserve bills in the vault itself.
Offshore Payday Edit
At the left side of the Crime.net screen there's an "Offshore Payday" icon. This lets you place a wager using large amounts of Offshore Money to get certain Card rewards. The cost of the Offshore Payday depends on the type and the number of "secured" (100% certain) cards that are going to be awarded to the player after this phase. The number of "Secured" cards can be none, up to a maximum of 3.
Note: the offshore payday is available only during a limited time after each DLC release. (usually disabled after a hotfix)