Why did Voldemort reveal himself at the end of OotP?

Question

During Order of the Phoenix, Voldemort tries his best to lure Harry to the Ministry so that Harry retrieves the prophecy for him.

In the aftermath, while Harry follows Bellatrix towards the exit, he raises a valid question:

”Why couldn’t he (Voldemort) come and get it (the prophecy) himself?”

This is a valid question, the prophecy is labeled as referring to the Dark Lord and maybe Harry Potter. Only those mentioned can retrieve the prophecy, so Voldemort should have been able to get it, assuming he is the Dark Lord of the prophecy.

Bellatrix gives a good answer:

“Get it himself?” shrieked Bellatrix on a cackle of mad laughter. “The Dark Lord, walk into the Ministry of Magic, when they are so sweetly ignoring his return? The Dark Lord, reveal himself to the Aurors, when at the moment they are wasting their time on my dear cousin?”

Despite the cackle of mad laughter, the reasoning is sound, Voldemort didn't want to reveal himself.

Yet he does exactly that when he appears in the Ministry and is seen by the Minister himself, and he exposes himself for no gain.

Bellatrix may or may not have been able to escape without his help, but he couldn't know that before he came, and his other followers that were captured were freed again a few months later, so it wouldn't have mattered much, he would have freed her, too.

So why did he go to the Ministry and reveal himself?

Answer 1

It would seem that Voldemort's decision to come to the Ministry was related to the loss of the prophecy. Note that when Harry is talking to Bellatrix at the beginning of Chapter Thirty-Six, he, Harry, 'feels' Voldemort's reaction to the prophecy being destroyed (my emphasis):

“Potter, I am going to give you one chance!” shouted Bellatrix. “Give me the prophecy — roll it out toward me now — and I may spare your life!”

“Well, you’re going to have to kill me, because it’s gone!” Harry roared — and as he shouted it, pain seared across his forehead. His scar was on fire again, and he felt a surge of fury that was quite unconnected with his own rage. “And he knows!” said Harry with a mad laugh to match Bellatrix’s own. “Your dear old mate Voldemort knows it’s gone! He’s not going to be happy with you, is he?”

Moments later Voldemort turns up:

“Don’t waste your breath!” yelled Harry, his eyes screwed up against the pain in his scar, now more terrible than ever. “He can’t hear you from here!”

“Can’t I, Potter?” said a high, cold voice.

Harry opened his eyes.

Tall, thin, and black-hooded, his terrible snakelike face white and gaunt, his scarlet, slit-pupiled eyes staring... Lord Voldemort had appeared in the middle of the hall, his wand pointing at Harry who stood frozen, quite unable to move.

This does not seem to be a coincidence. Apparently, upon learning of the prophecy's destruction Voldemort changed his plan. We can perhaps find a clue as to what this was from what follows:

“Be quiet, Bella,” said Voldemort dangerously. “I shall deal with you in a moment. Do you think I have entered the Ministry of Magic to hear your sniveling apologies?”

Here Voldemort says what is not the reason he came. This tells us that there is something else he came to do. And the very next thing he does is try to kill Harry:

“I have nothing more to say to you, Potter,” he said quietly. “You have irked me too often, for too long. AVADA KEDAVRA!”

So it seems that his new plan was simply to get rid of Harry once and for all. At this point his burning desire to eradicate Harry apparently overcame his caution. Additionally, he planned on doing a hit-and-run style attack, as is evident from his conversation with Dumbledore:

“It was foolish to come here tonight, Tom,” said Dumbledore calmly. “The Aurors are on their way —” “By which time I shall be gone, and you dead!” spat Voldemort. He sent another Killing Curse at Dumbledore but missed, instead hitting the security guards desk, which burst into flame.

Thus, he may not have been too worried about being spotted (and/or apprehended) in the few moments he planned on being there. And he may also have been been under the impression that the Death Eaters had secured the Ministry premises, so that might have removed some of his caution as well. Plus, he wouldn't necessarily have another easy opportunity to attack Harry.

Answer 2

The Dark Lord likely went out of his anger.

When the Dark Lord appears in the Ministry, he’s furious that Harry broke the prophecy and the Death Eaters didn’t stop him. He clearly didn’t plan on going, but he went once he realized his plan had been thwarted and he could no longer get the prophecy.

“He can’t hear you from here!’

‘Can’t I, Potter?’ said a high, cold voice. Harry opened his eyes.

Tall, thin and black-hooded, his terrible snakelike face white and gaunt, his scarlet, slit-pupilled eyes staring … Lord Voldemort had appeared in the middle of the hall, his wand pointing at Harry who stood frozen, quite unable to move.

‘So, you smashed my prophecy?’ said Voldemort softly, staring at Harry with those pitiless red eyes. ‘No, Bella, he is not lying … I see the truth looking at me from within his worthless mind … months of preparation, months of effort … and my Death Eaters have let Harry Potter thwart me again …”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 36 (The Only One He Ever Feared)}}

It seems most likely that he appeared at the Ministry out of anger, and didn’t have any logical reason for doing so - he most likely wanted to punish those responsible, and his fury overrode his sense. The prophecy was already smashed, and there would be less risky places to try killing Harry. However, he most likely wanted to punish the Death Eaters and kill Harry for causing his plan to fail, and his fury over the situation caused him to impulsively go to the Ministry. He similarly lashes out in anger when he discovered his Horcrux was taken from Gringotts.

“It was … it was … the P–Potter b–boy and t–two accomplices …’

‘And they took?’ he said, his voice rising, a terrible fear gripping him. ‘Tell me! What did they take?’

‘A … a s – small golden c – cup m – my Lord …’

The scream of rage, of denial, left him as if it were a stranger’s: he was crazed, frenzied, it could not be true, it was impossible, nobody had ever known: how was it possible that the boy could have discovered his secret?

The Elder Wand slashed through the air and green light erupted through the room, the kneeling goblin rolled over, dead, the watching wizards scattered before him, terrified: Bellatrix and Lucius Malfoy threw others behind them in their race for the door, and again and again his wand fell, and those who were left were slain, all of them, for bringing him this news, for hearing about the golden cup –”
_{^{- Harry Potter and the Deathly Hallows, Chapter 27 (The Final Hiding Place)}}

Though this insight into his mind shows he has no logical reason to kill the Gringotts goblins and those with them, he kills them out of anger at being told his Horcrux was missing. It seems like the situation at the Ministry was similar - he was furious at losing the prophecy and he went to the Ministry out of that fury.