The Directory was perceived of as corrupt and exhausted. Austria had declared war on France, and Napoleon was hugely popular -- the man the public thought could lead France in war.

Some of the more competent members of the Directory on good terms with Napoleon continued on into the Consulate and later the Empire-- the slippery diplomat Talleyrand and the brilliant mathematician and master of logistics Lazare Carnot, for example. Similarly, Joseph Fouché, a sometime ferocious revolutionary, got on fine with the new regime and served as Police Minister, occasionally hunting down his sometime revolutionary colleagues.

The coup d'etat that dispense with the Directory is called the "Coup Brumaire". Its organized by Talleyrand, Napoleon -- with substantial participation by his brother Lucien, and the Abbé Sieyès. Paul Barras a member of the Directory was part of the planning of the coup, but they got rid of him-- a healthy purse of cash and a suggestion that he retire to his country estate. He's the loudest in being pissed off about the affair; no one thought the Directory was working well, so apart from Barras and a few left over Jacobins there were few regrets. With Thermidor, the Revolution had been stripped of its zeal, and while the Directory was a great improvement on the reign of terror, France was at war and leader, people and the Army were all more confident in Napoleon's military acumen than the Directory's.

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Sources:

Memoires of Paul Barras -- self serving, but a lot of interesting detail. Not to be trusted without checking against another source

Memoires of Talleyrand-- same observations as the Memories of Barras. Useful, but remember you're reading what a very tricky man wants to tell you-- don't take it as gospel

Lazare Carnot, a forgotten piece in the Napoleon Bonaparte jigsaw

The Failure of the Liberal Republic in France, 1795-1799: The Road to Brumaire

“Brumaire: Bonaparte’s Justification,”