He had Ministry workers Imperiused, then had Scrimgeour killed.
The Dark Lord took over the Ministry mainly by having his Death Eaters use the Imperius Curse on Ministry officials to get them under his control.
“My Lord, I have good news on that score. I have – with difficulty, and after great effort – succeeded in placing an Imperius Curse upon Pius Thicknesse.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
Their plan for Scrimgeour, though, was always to have him killed - they didn’t expect to control him. Instead, they planned to Imperius enough officials around him so they could successfully kill him.
“It is a start,’ said Voldemort. ‘But Thicknesse is only one man. Scrimgeour must be surrounded by our people before I act. One failed attempt on the Minister’s life will set me back a long way.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
They planned to use the officials they already had under the Imperius Curse to Imperius or otherwise control the others, and once they got enough of the Ministry officials converted, they’d use them to kill Scrimgeour. Though he was aware, he was still only one man, so could be overpowered by the others, despite his knowledge that the Dark Lord was trying to take over.
“Yes – my Lord, that is true – but you know, as Head of the Department of Magical Law Enforcement, Thicknesse has regular contact not only with the Minister himself, but also with the Heads of all the other Ministry departments. It will, I think, be easy, now that we have such a high-ranking official under our control, to subjugate the others, and then they can all work together to bring Scrimgeour down.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
This is how the Dark Lord successfully gained control of the Ministry - by putting the Imperius Curse on officials who were below Scrimgeour in rank, and using them to kill him.