Note that $E[X|Y]$ is a function of $y$. $$\begin{array}{rl}E[E[X|Y]]& =\sum _{y}E[X|Y]p_Y(y)\\ & =\sum _y\sum _{x}x{p}_{X|Y}{p}_Y\\ & =\sum _y\sum _{x}x{p}_{X,Y}(x,y)\\ & =\sum _{x}x\sum _y{p}_{X,Y}(x,y)\\ & =\sum _{x}x{p}_X(x)\\ & =E[X]\end{array}$$