A bridge is a spanning member (tube) that connects (usually) a pair of seatstays or chainstays. Bikes with rim caliper brakes need a place to hang the brake, and bikes with cantilevered brakes need something to resist the [non-frictional] forces of braking. Bicycle brakes work by applying an inboard force against some part of the wheel, be it a rim or a rotor. The inside of the hubshell is pressed against in the case of a coaster brake. This [normal] force results in a frictional force between the braking surface and the braking media that opposes the rotation of the wheel, slowing it down. In addition to the frictional force that results from this braking force, a counterforce is exerted, by the brake surface, on the brakes. This force is in the opposite direction (thanks Newton) of the applied force from the brake. Since the brake itself is fairly rigid, this force is transmitted to the mounting point on the frame. Without a bridge to hold those mounting points (on the stays) in a relatively rigid location, the stays would bend outward. This bending uses some of the force that the rider has put into braking and the remaining [normal] force (which determines the amount of friction between the braking surface and media) will be reduced. With less friction, there is less braking. A bridge resists this bending with a tension force, allowing the maximum amount of friction to be generated with the braking input force.
Disc brakes are similar to road rim calipers in that they take care of all of their own lateral force balancing and only require that the longitudinal forces (those that arise from the acceleration of the wheel) be balanced. As the brakes are applied, frictional forces are 'pulling' the caliper in the direction of wheel rotation. If the caliper is mounted on a seatstay, a strut (bridge) is installed to the chainstay to brace against a larger tube (seatstays < chainstays). If the caliper is mounted 'low', the chainstay will do the bulk of the resistance work without the need for extra bracing. Ovalized chainstays are now the norm, since they offer good resistance to not only the forces applied during pedaling, but also this 'vertical' force of braking.
So where does this leave the bridges that span the stays out past the outer reaches of the tire?
Vestigial ornaments of a bygone era.
Do they add 'strength' and stiffness to the frame still? Empirically. Is that difference appreciable in practical application? Well, that's certainly the question isn't it? Over-engineering is great, but adding unnecessary structural elements to a machine where weight-savings are counted in grams can be dubious. When compared to the 'classic' road machine: A lugged steel frame with 1" and 1-1/8" diameter tubing; modern bikes are appreciably stronger (and stiffer) by virtue of improved metallurgy and increased tubing diameters and profiles. Short, over-sized, ovalized chainstays; an oversized thru axle; and the intrinsic stability of the rear 'triangle' (a tetrahedron), all add up to a compelling argument for removing this archaic strengthening member.
The forces of pedaling exert large torsional forces on the lower region of the frame. The bottom bracket shell is pushed away from the side of the bike that's receiving the greatest input from the rider. This twists the downtube, bends the seattube, and skews the triangle of the chainstays. Bridges are placed between the chainstays to fight this distortion. Does this work? Maybe. Is it noticeable? Again, probably not. Would the frame receive the same or better resistance to this distortion by laterally ovalizing the seattube and possibly the ends of the chainstays as they approach the bottom bracket? Possibly. If so, they'd certainly do it without adding weight to the bike.
Well, that brings us to the end of this little note. What'd we learn? As
with any craft, each decision, each piece of the whole should be
consciously considered by the craftsperson. While many details can fall
both into the 'form' and 'function' categories, care should be taken to
not mis-categorize items based purely on historical precedent.
As a rule, I don't install stay bridges on any of the frames I build. I will install one if I see that the shape of the stays has created a natural void that just begs for an additional curve. Ornamentation.