In general, to prove these types of inequalities you want to work backward (i.e., simplify what you're given) until you get something you know is true, then work forward for your proof. For example, getting rid of the fraction and squaring both sides, then combining terms and factoring gives something true...
add $4xy$ to both sides,
multiply both sides by $xy$ to get
then divide $(x+y)^2$ over and take the square root to get
Of course equality holds when $x=y$.