suppose
y = b0 + b1*x1 + b2*x2 + error
If we impose the restriction b1 = 1, then just run the transformed regression
y-x1 = b0 + b2*x2 + error
In other words, replace the original dependent variable y with y-x1
If we impose b1 + b2 = 1, then
y = b0 + (1-b2)*x1 + b2*x2 + error
Rearrange, we get
y-x1 = b0 + b2*(x2-x1) + error
So the new dependent variable is y - x1, and new independent variable is x2-x1.