In the RoTK movie Gandalf even says "Sauron will suspect a trap".Even if the thought of someone trying to destroy the ring doesn't ever cross Sauron's mind, there was no need to immediately respond to the challenge.Especially since this happened immediately after his defeat at Pelennor fields.There was no way the much smaller army of men could have defeated Mordor in open war.
In the movie as released in the theaters, if I'm remembering correctly, this is a huge unexplained element of the ending.
In the extended edition, we get a bit more explanation, in the form of an extra scene with Aragorn prior to the final battle. In this scene, while they are recovering from Pelennor Fields, Aragorn uses Minas Tirith's Palantir to challange Sauron, and shows him Anduril (the reformed Narsil)
The idea behind this scene is to do several things:
- Taunt Sauron with his recent defeat and threaten to continue beating Mordor's armies
- Remind Sauron of the last time he was defeated, at the hands of a Dunadan wielding Narsil, and that the descendant of that Dunadan wielding basically that same sword was coming after him again
- Convince Sauron that Aragorn had The One Ring.
The idea here is that, the last time Sauron definitively knew where the Ring was, Frodo had just used it at Amon Hen, while he was still in the company of the Fellowship. We can also assume that Saruman was keeping him appraised of the goings-on there, and that Aragorn was one of those people with Frodo. Given that the Ring was never used again, and that the few times the Orcs did capture a Hobbit they were ring-free, Sauron had no way of knowing if a "Baggins" still had it.
Of all of the people on Middle Earth at that time, only a handful of them would have been considered even remotely a threat to Sauron; of those, Sauron probably recognized that elves like Galadriel, or Maiar like Gandalf, would refuse to put on The Ring for fear of it's corruption. That left one person: Aragorn, descendant of Isildur, who might actually chance it.
To someone like Sauron, given that the Ring was a powerful weapon, he would assume they would give it to the most powerful person around, and that person would want to use it. So, it makes sense that he would buy that Aragorn had it. Indeed, to Sauron, the simple fact that Aragorn dared threaten him, knowing how powerful Sauron and his armies were, was likely more proof that he must have the Ring. So, Sauron sent his armies out on the premise that they would do one of two things:
- Capture Aragorn and get the Ring back, or
- Drive Aragorn to enough desperation to put the Ring on, whereby Sauron could try to influence him.
He didn't want to give the humans time to reconsider their plan, or come up with a better one, or risk letting them figure out a way to use the Ring against him. Plus, he's just greedy, and he wanted the Ring back. It was worth more to him then his entire army of Orcs and Uruks combined.
My main issue with the way this played out in the movies is that Sauron knew there was at least one Hobbit in Morder, and he knew that Hobbit had escaped. In the Extended Edition, the Mouth of Sauron taunts Aragorn with the mithril shirt Frodo was wearing. Sauron also presumably knew that, at least at some point, that a Hobbit was carrying the Ring around.
Thus, as much as Sauron might want to assume that Aragorn had it, and as much sense as that would make, I find it stretches credibility that he found a Hobbit loose deep inside Mordor, and it didn't set off immediate alarm bells to watch Mount Doom more closely.
However, credibility issues aside, that is the intended explanation for why Sauron was so willing to met Aragorn's challenge so quickly.
Based on the Peter Jackson movie commentary, Sauron's primary motivation for opening the Black Gates in the movie adaptation was that he received a personal challenge from Aragorn. In the extended cut, we see Aragorn using the Palantir to taunt Sauron, showing him Andúril reforged and demanding that he face him. It's worth noting that in the original cut, Sauron did indeed ride out to face Aragorn himself (in a one-on-one encounter) rather than just sending his troops.
Peter Jackson : "...this is how we originally conceived the sequence, which was to be challenging Sauron which is obviously... it's irrelevant...but more relevant to some degree to the fact that when we shot this we had already filmed Aragorn and Sauron actually fighting outside the Black Gates so this was the personal challenge that was going to bring Sauron to the Gate.
... it still works [in the theatrical cut] because he's basically challenging Sauron's armies to come out to meet them, which is fine.