Simplify or rules don't work with square roots

You can simplify using Surd[ ,2] instead of Sqrt

Sin[Surd[x + y, 2]]/Surd[x + y, 2] /. Surd[x + y, 2] -> z
Sin[z]/z

Another option is to help it a little

ClearAll[x, y];
expr = Sin[Sqrt[x + y]]/Sqrt[x + y]

expr /. {1/Sqrt[x + y] -> 1/z, Sqrt[x + y] -> z}

Yet an other way:

sol = First@Solve[Sqrt[x + y] == z, x]

      (*   {x -> -y + z^2}   *)

Sin[Sqrt[x + y]]/Sqrt[x + y] /. sol // 
    PowerExpand[#, Assumptions -> z > 0] &

Or

Sin[Sqrt[x + y]]/Sqrt[x + y] /. sol // 
     Simplify[#, Assumptions -> z > 0] &

(*   Sin[z]/z   *)