How can Dr. Strange use the Time Stone to do this?

I can't back this up or anything, but I think it has to do with the fact that Strange can use the Time stone, while the Ancient One can't.

The Ancient One was powerful, but she just guarded the stone in the sanctuary, she didn't/couldn't use it while Strange could. So I'm guessing that the Time stone is the key here and it could probably grant Strange insight in the future further from the moment of his death.

The other answer is legit and could be right, but I have a different take on this.

The Ancient one said to Dr. Strange in that same conversation that she saw his future, or more accurately, his possibilities. She also said that only with Mordo that he stood a chance against Kaecilius/Dormammu. Those two statements strongly suggest to me that she could see past that moment in other people's futures, she just can't see her own future past that moment, despite looking many times, perhaps, (just maybe), 14,000,605 times. ;-)

It's unclear whether Dr. Strange really had any skills that the Ancient One didn't. He may not have been able to view past his own death either, though his statement to Tony "There was no other way", suggests that he did see past that moment in other people's futures. So I would say the skills are the same, not different because of the time stone.

But only the writers know for sure. :-)

Does this mean the wielder of the Time Stone can't see past their own death?

We cannot be sure but it seems a logical assumption to make, based on the fact that the Ancient One is not able to see past her death. Although we do not know if Dr Strange can, we have not been given any reason to think that the rules are different for him.

Can Dr. Strange do what the Ancient One couldn't: see beyond his own death?

No, and yes.

I think it is logical to think that the only reason Dr Strange is able to see past his own death, is because he didn't stay dead.

Think of this as an example: it is not clear if Dr Strange actually died on the operating table for a brief period in his origin film, before being revived. However, let's assume for a moment that he did - would you expect that he would not be able to see the future past that point? No, of course not, because he still had a future to see.

It is not the momentary act of dying which prevents the stone wielder from seeing their future, but instead, it is the eternal cessation of living.

When he looked at every possible outcome, the ones where he could not see anything after his death on Titan, he would chalk up as losses, because it meant that he was never revived.

A Note On @Plexus Answer

@Plexus answer asserts that the Ancient One cannot use the Time Stone. While this is possible, I think it's unlikely. It is clear that the Ancient One can see into the future and manipulate time. We are not given any reason to assume that she uses any magic other than the Time Stone to do that.

Also, Banner goes to collect the Time Stone from the Ancient One, she is wearing it - presumably so that she can use its powers if she needs to. You could argue that she was wearing to ensure its protection. However, if that were the case, it would be logical to think that she would always keep it with her. However, when Dr Strange is studying, it is kept in the Sanctum, and Wong acts as though it is always kept there. That leads me to believe that it was normally kept in the Sanctum except when the Ancient One might need to use it.