A more thorough explanation:

A veto-proof majority is a legislative majority that is large enough to override an executive veto. This means that if the executive (such as a president or governor) vetoes a bill, the legislative body can still pass the bill into law without the executive's approval.

For example, let's say that a state legislature passes a bill with a vote of 40-10. If the governor of that state vetoes the bill, the legislature would need a veto-proof majority of at least 2/3 (or 27 votes) to override the veto and pass the bill into law.

Another example is the United States Congress. In order to override a presidential veto, both the House of Representatives and the Senate must pass the bill again with a 2/3 majority in each chamber.