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