Algebraic simplifications like Simplify and FullSimplify can be used with the second argument - assumptions. We can assume e.g. that b is a real number i.e. b ∈ Reals :

Simplify[ (1/(b (-1 + E^b) Re[b])) E^-Re[b](-b + b E^b + E^Re[b] Re[b]
           - E^(b + Re[b]) Re[b] + E^Re[b] Sqrt[E^(-2 b) (-1 + E^b)^2] Re[b]
           + b E^(b + Re[b]) Sqrt[E^(-2 b) (-1 + E^b)^2] Re[b]),  b ∈ Reals] // 
 TraditionalForm

Expand @ %[[2]] // TraditionalForm

The same answer you can get with FullSimplify.

Algebraic simplifications like Simplify and FullSimplify can be used with the second argument - assumptions. We can assume e.g. that b is a real number i.e. b ∈ Reals (otherwise the system assumes that b is complex) :

Simplify[ (1/(b (-1 + E^b) Re[b])) E^-Re[b](-b + b E^b + E^Re[b] Re[b]
           - E^(b + Re[b]) Re[b] + E^Re[b] Sqrt[E^(-2 b) (-1 + E^b)^2] Re[b]
           + b E^(b + Re[b]) Sqrt[E^(-2 b) (-1 + E^b)^2] Re[b]),  b ∈ Reals] // 
 TraditionalForm

Since there are two cases b >= 0 and b < 0 (in general there might be more cases depending on the assumptions) we should map Expand on the output ( common shorthands Map -- /@ and MapAll -- //@)

Expand //@ % // TraditionalForm

The same answer you can get with FullSimplify.