Each individual proxy fight has the potential to be unique, but most proxy fights follow a common thread. The typical way that a proxy fight works is that shareholder activists are dissatisfied with a particular aspect of the company and seek change in that area; however, they often run into resistance from the company's current board members. The dissatisfied shareholders then attempt to persuade other shareholders to allow them to use their proxy votes on a proposed change to the company's board positions.
The shareholder activists typically attempt to remove board members that oppose their desired changes and install their own board member candidates. The new board members will be receptive to the changes proposed by the shareholder activists, making it easier for the activists to make those changes happen. (See also: How Your Vote Can Change Corporate Policy and Proxy Voting Gives Fund Shareholders a Say.)