Becuase of his past relationship to Voldemort and his skill with Potions
While Harry’s story was widely dismissed by both the Ministry of Magic and the wizarding press, Horace Slughorn believed it. Confirmation came three nights after the death of Cedric Diggory, when the Death Eater Corban Yaxley arrived at Slughorn’s house under cover of night, clearly intending to recruit him, or take him by force to Voldemort.
Slughorn reacted with a speed that would have astounded those who had watched him grow slower and fatter through the years of his retirement. Transfiguring himself into an armchair, he successfully evaded Yaxley’s detection. Once the Death Eater had left, Slughorn packed a few necessities into a bag, locked up his house behind him, and went on the run.
For over a year, Slughorn moved from house to house, often squatting in Muggle dwellings when the owners were away, because he did not dare stay with friends who might subsequently betray – whether willingly or under duress – his whereabouts. It was a miserable existence, made still more wretched by the fact that he did not know precisely what Voldemort wanted from him. He thought it most likely that his old student simply wanted to recruit him to his army, which was still small compared to what it had been at the height of his previous power; in his darkest moments, however, Slughorn wondered whether Voldemort did want to kill him, to prevent him ever betraying the source of the latter’s continuing invulnerability.
Pottermore - Horace Slughorn (behind paywall)
While Slughorn briefly thought Voldemort wanted to kill him, this was later revealed to be false.
Once the school had been taken over by Lord Voldemort, with Severus Snape as Headmaster and the Death Eater Carrows taking key roles in subjugating staff and pupils, Slughorn learned that Voldemort had nothing worse in store for him than to remain in post and teach pure- and half-bloods. This he did, keeping his profile as low as he dared, though never enforcing the violent discipline advocated by the Carrows, and attempting to look after the students in his care as best he could.
Pottermore - Horace Slughorn (behind paywall)