He seemed to believe that Draco's plans were doomed to failure.
The question is right to say that Dumbledore didn't take Harry's warning seriously. The protection from the Order of the Phoenix which amflare describes had been arranged before that conversation. Dumbledore did take additional precautions before leaving with Harry but, as the quote in the question demonstrates, he did this every time he left the school that year.
Dumbledore did not expect an attack of any sort that evening.
"...yes, you have managed to introduce Death Eaters into my school which, I admit, I thought impossible...how did you do it?"
(Half-Blood Prince, Chapter 27, The Lightning-Struck Tower).
He had introduced protective spells over the summer holidays and asked the Order to guard the school when he wasn't there, but this was generic protection. It couldn't be clearer from his words and his reaction that the Malfoy attack took him completely by surprise.
So why didn't he take Harry seriously?
Out-of-universe:
From a literary perspective, this is just tragic irony. Dumbledore's death was (on the face of it, from the point of view of a pre-Deathly Hallows reader) entirely preventable. If only Dumbledore had listened to Harry it could all have gone so differently. The same thing could be said of the other deaths in the series. What if Harry hadn't let Cedric touch the Triwizard Cup? What if Harry hadn't believed the fake vision of Sirius? And so on and so on. It just makes Dumbledore's death sadder and more pitiful.
In-universe:
Dumbledore seemed to conclude that Draco's attempts at killing him were doomed to failure. This turned out to be a fatal error of judgement. But Dumbledore underestimated Draco, pure and simple.
After all, Draco's efforts up to that point had hardly been exemplary.
"You have been trying, with increasing desperation, to kill me all year. Forgive me, Draco, but they have been feeble attempts...so feeble, to be honest, that I wonder whether your heart has been really in it..."
(Half-Blood Prince, Chapter 27, The Lightning-Struck Tower).
Dumbledore was more worried about Draco and any unintended victims of his plots than he was about himself.
"Your first priority will be to discover what Draco is up to. A frightened teenage boy is a danger to others as well as to himself. Offer him help and guidance, he ought to accept, he likes you...I am concerned less for myself than for accidental victims of whatever schemes might occur to the boy."
(Deathly Hallows, Chapter 33, The Prince's Tale).
The fact is that Dumbledore had enough faith in his own abilities that he didn't consider Malfoy a true threat. He had no way of knowing that he would return to the school under the influence of a dreadful potion which would weaken him considerably. He believed that he'd be able to control Malfoy in any direct confrontation and certainly prevent Death Eaters from wreaking havoc in the school in the way in which they did. He knew what he was capable of and how powerful he was. He's Dumbledore, remember. He's used to getting his way in any fight he finds himself in, and in being able to protect the people he cares about when he's around.
He was worried more for Malfoy and the other Hogwarts students than he was for himself. As far as he was concerned, he'd put Snape in charge of monitoring Malfoy and the Order were going to be around, so the school was safe. Any threat from Malfoy wasn't going to be a real cause for concern.
Why not go down to the room and check on the off-chance? Well, Harry's warning wasn't exactly delivered in the most effective way. He had just discovered that Snape directly betrayed his parents (resulting in their deaths) and that Dumbledore had never told him. This caused him to fly off in a rage. Harry was furious, irrational and full of hatred. His warning about Malfoy was mixed in with a rant against Snape.
At last [Dumbledore] said, "I am sure. I trust Severus Snape completely."
Harry breathed deeply for a few moments in an effort to steady himself. It did not work.
"Well, I don't!" he said, as loudly as before. "He's up to something with Draco Malfoy right now, right under your nose, and you still -"
"We have discussed this, Harry," said Dumbledore, and now he sounded stern again. "I have told you my views."
(Half-Blood Prince, Chapter 25, The Seer Overheard).
That Harry had mixed up a legitimate concern with his rambling personal grievances against Malfoy and Snape didn't help. Dumbledore also knows that Harry has already come to him with this concern before, and he didn't inform him of anything he wasn't already aware of. Dumbledore thought they were covering old ground.
He did go as far as trying to ascertain what exactly Harry was alleging, though.
"You're leaving the school tonight and I'll bet you haven't even considered that Snape and Malfoy might decide to -"
"To what?" asked Dumbledore, his eyebrows raised. "What is it that you suspect them of doing, precisely?"
"I...they're up to something!" said Harry, and his hands curled into fists as he said it. "Professor Trelawney was just in the Room of Requirement, trying to hide her sherry bottles, and she heard Malfoy whooping, celebrating! He's trying to mend something dangerous in there and if you ask me he's fixed it at last and you're about to just walk out of school without -"
"Enough," said Dumbledore. He said it quite calmly, and yet Harry fell silent at once; he knew that he had finally crossed some invisible line.
(Half-Blood Prince, Chapter 25, The Seer Overheard).
So he gives Harry a chance to explain himself. The fact that he asks "with his eyebrows raised" I think indicates that he is not exactly asking with an open mind. He's trying to diffuse Harry's anger, and demonstrate that he's being irrational. He doesn't expect Harry to tell him anything meaningful. But he does at least give Harry the space in which to tell him his suspicions.
As I say, Harry doesn't help himself. When given the chance to tell Dumbledore the truth he just says that Malfoy and Snape are up to "something". Immediately, it sounds as if he's clutching at straws. The only concrete detail in his story is that Professor Trelawney was thrown out of the Room of Requirement by a gleeful figure. Yet why should this be Malfoy? How can Harry know that he's trying to fix something dangerous? His accusations descend quickly into wild speculation ("if you ask me"). No wonder Dumbledore says that he's heard enough.
An additional factor may be that Dumbledore himself has become angry. He doesn't intervene until Harry suggests that Dumbledore is leaving the school unattended. This provokes a stern response. Dumbledore has the patience to listen to Harry's story but snaps at the suggestion that he doesn't care about the welfare of his students. This, after all, is something that is close to Dumbledore's heart.
Of course, even if he had gone down to the Room of Requirement he would've been confronted with the same thing as Harry had been all year - a locked door with no means of entry.
Finally, Dumbedore isn't the only one to blame here. Harry too heard a clear warning of what was to come and shrugged it off. Trelawney shared her premonition about the lightning-struck tower with Harry and he thought that she was talking nonsense when she was actually seeing the future.
