With help.
Although the clones were one of the most visible symbols of the Republic and were critical in Palpatine's overall plan, they weren't the only forces fighting for the Republic. Two other key elements of the Republic's defense were the Judicial Forces and the Planetary Defense Force.
The Judicial Forces were primarily naval forces, suited to patrolling hyperspace routes and trade lanes, or raiding Confederacy shipping. Planetary Defense Forces, naturally enough, were raised by individual worlds in the Republic and were intended for defensive use, but could be deployed on campaign if absolutely necessary.
Because these forces were raised at the local level across the Republic, they were extremely numerous but not very centralized. Their roles were therefore to protect and hold Republic territory while the clones of the Grand Army carried out more vital operations that required greater coordination and concentration of force.
When the Empire was declared, these forces along with the clone army were folded into the new Imperial Army and Navy; many non-clone Imperials were originally Judicial Forces or Planetary Defense troops. The eponymous officer of the Tarkin novel (in Canon) and Gilad Pellaeon (in various Legends works) were notable examples of Imperial officers who once fought for the Republic.
It's worth considering, too, that neither side was really interested in large-scale invasions for their own sake. They both believed that if they removed the enemy's leadership, the other systems would more or less fall in line, so they were more concerned about attacking specific, vital targets than in occupying every square kilometer of the other side's planets.
