TdF: How does putting one team member in a break away help the rest of the team? In a race such as Le Tour De France, what advantage does a team gain from sending one team member into a break away? What does the rest of the team, including their favoured rider, gain?
The full answer depends exactly on the teams involved and the composition of the break but, in general, the key to understanding this isn't physics as much as it is game theory. 

In a race like the TdF, there are many separate competitions being run simultaneously: not just for the overall General Classification but also for stage wins, the (green) points jersey, the (polka dot) mountain jersey, the (highest ranked rider under age 25) white jersey, the team with the lowest cumulative time, and so on. Not every team has a GC hopeful, but every team has a goal or goals. If your goal is a stage win, or to get one of your riders a particular jersey, getting someone into the break can allow the rest of the team to shelter in the peloton, conserve their effort for the entire day, and prepare for the next day's stage. 

A more complicated case is when you do have a GC contender and also someone in the break. Depending on the composition of the break, your break rider can be threatening some other team's goal of a jersey. In that situation, you're playing a waiting game (a game of "chicken"), to force the threatened team to chase the break, while your team shelters in the peloton, conserving their effort for the entire day. 

Sometimes a GC-contending team will put a rider in an early break with a set plan to drop the rider out of the break later in the race to help the team's leader. It's like caching supplies farther up the road that they can draw upon if needed. 

Sometimes you have a GC contender but no one in the break, and the break composition threatens the secondary goal of another team. If so, you can wait until the other team starts to panic and initiates the chase, then shelter in the peloton, conserving your effort. 

Teams will always look at the riders of an early break to determine whether its composition fits their goals. If not, the team will chase early to absorb the break, then wait until a different, more acceptable, break forms; in which case they will allow the break to continue while they shelter in the peloton and wait for someone else to panic and initiate a chase. 

As the stages count down in one of the Grand Tours (the Tour de France, the Giro d'Italia, or the Vuelta a Espana), teams that still haven't yet won a stage become more and more desperate to get someone into an early break. There can be several re-shufflings of break compositions as the directeur sportifs try to get at least one rider into a stable break. That can be miserable for already tired riders as they accelerate again and again to claw back unsuitable breaks.  

There are 20 teams on the road simultaneously each with different goals and skill sets, with many different sub-competitions. In terms of n-person game theory, cycling allows (encourages) temporary coalitions, collusion (as when members of different teams in a break work together to stay away), side payments (not only of money but also of agreements not to contest a sprint or mountain point), prisoner dilemmas, bluffs, feints, and tops it off with reneging. Then they have to do it again the next day, remembering whether a particular team carries through with its promises. This is the reason why mass start bicycle racing is, from both a strategic and tactical perspective, so much more complex than sports that feature only two teams facing each other on a field.