The ++ operator increments the value and returns the original value. 
So when x was 5, x was incremented and the condition 5 > 5 is evaluated.
Then, when x was 6, x was incremented and the condition 6 > 5 is evaluated, which results in the break. Because x was still incremented, the value of x in the end is 7.
People often say that when the ++ operator is use as a postfix, the increment is executed after the comparison, but this is not technically true, which is shown by decompilation in LinqPad:
int x = 5;
if (x++ > 5) 
Console.WriteLine(x);
IL:
IL_0001:  ldc.i4.5    
IL_0002:  stloc.0     // x
IL_0003:  ldloc.0     // x
IL_0004:  dup         
IL_0005:  ldc.i4.1    
IL_0006:  add         // <---- Increment
IL_0007:  stloc.0     // x <-- Result of increment is popped from the stack and stored in a variable
IL_0008:  ldc.i4.5    
IL_0009:  cgt         // <---- Comparison
IL_000B:  ldc.i4.0    
IL_000C:  ceq         
IL_000E:  stloc.1     // CS$4$0000
IL_000F:  ldloc.1     // CS$4$0000
IL_0010:  brtrue.s    IL_0019
IL_0012:  ldloc.0     // x
IL_0013:  call        System.Console.WriteLine