To find out how many files are in a board, look at the board summary included with any board view. The board summary provides a quick glance at the board contents, showing a total sub-board count, total asset count, total file count, and total storage included.
Keep in mind that the storage included in a board summary is not always indicative of storage in use across the workspace, as files added to your workspace can appear in many boards or no boards at all. To see your total current storage in use, head to your workspace settings.
If a board has several sub-boards, use the 'Expand nested items' button to update the board summary on the board view footer, allowing you to get a count of all assets across the board and all sub-boards.