Equivalence means reflexive, symmetric and transitive. I think you know their definitions.
reflexive part: (to show that xRx holds) :xRx holds since x+x=2x is even.
symmetric part (to show that if xRy holds then yRx also holds): if xRy holds then it is implied that x+y is even, that is y+x is even, that is yRx also holds.
transitive part (to show that is xRy and yRz holds then xRz also holds): if xRy and yRz holds then it is implied that x+y is even and y+z is even; again y+z even means z-y is also even; then x+y+z-y is also even, that is x+z is even; which means xRz also holds.