At the beginning of Revenge of the Sith, we see a massive battle over Coruscant. It's pretty clear that this is the height of the Clone Wars.
So how were the Separatists able to get so deep into Republic territory?
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According to the (canon) Star Wars: Battles That Changed the Galaxy factbook, the Separatists took advantage of two key elements, that the defensive perimeter around the core planets was relatively sparse and their possession of a secret hyperspace route from Separatist-controlled space into the Coruscantii system.
It was at this point that Count Dooku and General Grievous staged the audacious—and potentially war-winning—raid on Coruscant, the galactic capital. The Republic’s most battle hardened forces had been drawn to the Outer Rim, leaving the interior relatively undefended. Utilizing secret hyperspace routes through the Galactic Core, the Separatist fleet was able to arrive at Coruscant without warning. With the Coruscant Home Defense Fleet outmaneuvered, General Grievous led an expedition to the Senate District, took Supreme Chancellor Palpatine hostage, and returned to orbit.
As the film's official novelisation says, this wasn't a proper invasion force, it was a "lightning raid" conducted deep into Republic territory. The aim here was to capture the Chancellor whatever the cost in lives and equipment to the Separatists. Within minutes of their arrival, their (small number of) capital ships were under heavy attack and the assault troops they'd landed were escaping back into space:
We won! beings tell each other. We held them off!
But then new reports trickle in - only rumors at first - that the attack wasn’t an invasion at all. That the Separatists weren’t trying to take the planet. That this was a lightning raid on the Senate itself.
The nightmare gets worse: the Supreme Chancellor is missing.
Note that this attack mirrors real world attempts to decapitate the enemy during warfare.
The Junior Novelisation also gives a similar account (and a bit more info) in its prologue
Then the Separatists struck a paralyzing blow, straight at the heart of the Republic. A fleet of ships commanded by the dreaded Separatist General Grievous slipped through the outer line of defenses to attack Coruscant itself. In the confusion the Separatists kidnapped Supreme Chancellor Palpatine, the elected leader of the Republic.
But Coruscant was not only the heart of the government and the location of the Galactic Senate. It was also the home of the Jedi Temple. As the Separatist attack began, a message was beamed to the Outer Rim, summoning the Jedi’s greatest warriors home. Before the Separatist fleet could leave the Coruscant star system with the Chancellor, they found themselves under attack. Waves of clone starfighters, led by Obi-Wan and Anakin, stormed around their ships….
Palpatine was playing both sides as Chancellor of the Republic and Sith Lord (leading the Separatists). From Star Wars wikia on the Battle of Coruscant:
Through the use of a secret hyperspace passage through the Deep Core supplied to them by Darth Sidious, the Separatists managed to jump right on top of the Coruscant Home Fleet, catching them completely by surprise.
During the Clone Wars, Supreme Chancellor Palpatine dispatched scouts and probe droids to explore the region. As the Sith Lord Darth Sidious, he provided the Separatist leadership with access to secret hyperspace routes that allowed them to bypass the heavily-defended Corellian sector and attack Coruscant in 19 BBY.
Essentially, Palpatine gave them a key to the back door while the Republic forces were only watching the front door
During the Clone Wars, Darth Sidious sent many probe droids into the Deep Core- the region which contained the galactic center. The Deep Core contained stars and planets so close together, that going into hyperspace was often dangerous.
The probes however, found a few safe routes in the Deep Core, which the Separatists used to reach Coruscant