The phrase "big check" means a substantial amount of money, usually in the form of a large sum of money being given or received for something important or significant.
It didn't take long for the special interests to start writing big checks. (nypost.com)
Not anymore: All the major car companies have joined the fray — and shown a willingness to cut big checks for the tech behind it. (recode.net)
If such a system would have been implemented, it could have enabled candidates who can't garner big checks from special interests to compete with small dollar donations from constituents. (reinventalbany.org)