The management philosophy of a firm is the same thing as its managment style. It is the set of ideas that governs how a firm will attempt to manage its employees.
For example, in the past, management philosophy tended to be rather authoritarian. In a firm, higher level managers would give orders and those below would follow them. There was no attempt to try to show the lower level employees that their ideas were valued or to try to build a team spirit among workers. Today, many firms have moved away from this management philosophy. They now opt for philosophies that emphasize making their employees feel like valued members of a team. In both cases, the management philosophy is a basic set of core values about how the business should be run. These values or philosopies dictate how the firms attempt to manage their employees.