Bridge is the coordinating hub of the Creative Suite. Synchronizing color management settings for all suite programs is done from Bridge, and can only be done from Bridge, to take one important use.
As Lauren mentioned, Bridge displays actual thumbnails of many more file types than Finder or Explorer. It also allows instant play of sound or video files more readily than the native OS file managers.
Bridge allows direct access to file metadata, to embed copyright information and keywords where appropriate (e.g., for corporate logo vector and raster files). It also displays the fonts used in an InDesign file, the swatches in an INDD or AI and the output plates (including spot color plates) they use.
When managing the assets for a design project, Bridge allows quick and simple sorting, rating and custom labeling (with color flash indications) of assets. I can rate images according to whether they are rejects, possibles, for review by client, or approved. The filters built into Bridge allow instant isolation of only the approved images or designs in a folder, only the rejects (for deletion) or only files with certain ratings, no matter how many files it contains. It recognizes aspect ratios, so if I only need a landscape or a 16:9 image in a folder of hundreds of images, I turn off the aspect ratios I don't need.
Once filtered, the remaining visible files can be selected and copied, moved, or deleted without affecting the rest of the contents of a folder.
Collections are a massively useful feature. One of my clients is a performing arts center, and in a season we turn out dozens of ads, flyers, brochures, web banners, playbills, billboards and other collateral using the same assets over and over. These assets are organized by artist and/or show on disk, but I set up each season's repeating assets as a Collection in Bridge, so that I just have to open the collection and drag and drop these assets into new INDD, AI, PSD, HTML (in Dreamweaver), FLA or AE projects without having to navigate from folder to folder picking up individual files.
Bridge's Favorites is another place I stack frequently-accessed folders, such as stock photography, backgrounds, and top-level folders for active projects.
Assets can be divided into subfolders, but a quick toggle of "Show items from subfolders" exposes all of the assets in a single view while maintaining their organization. I will typically keep AIs, PSDs, EPSs, stock photography and client images in separate subfolders within a project. When I'm ready to start pulling assets into an InDesign layout, I toggle this on and simply drag what I need into the layout.
Bridge comes with Adobe Camera Raw built in, which is many times faster than using Photoshop to adjust jpegs or tiffs for things like tonal range, white balance, cropping, spotting and sharpening, and is non-destructive.
One tremendously useful Bridge function for InDesign CS5+ users is the "Show linked files" feature, which opens all the linked files in a layout into a single view, regardless of where they are physically located. I often use this when doing alternative layouts from a client-approved mockup for a campaign, to be certain the same assets are used in each piece, or when creating a motion graphic or interactive piece for the campaign in After Effects or Flash.
The batch and image processing scripts built into Bridge automate things like creating web-ready small jpegs from multiple images, renaming large numbers of files in place or by copying to an alternative location, creating sets of PSD, png, jpeg or other file types from an assortment of image files, and so on.
Bridge is so much a part of my daily workflow that on my main workstation I have one monitor dedicated to it almost 100%. Bridge just sits open 24/7, ready for use. I would run at half speed without it, no question.