You're confusing "transwarp" (TNG/VOY) with "multiwarp" (TOS/TAS).
In the original Star Trek, the maximum velocity the Enterprise was capable of travelling safely at was Warp 10 but this didn't mean that speeds faster were impossible to reach.
In several episodes of TOS, the Enterprise and various other items travel faster than warp 10 both with and without alien assistance;
Journey to Babel - The Enterprise encounters a ship capable of travelling 'faster than Warp 10'
Let That Be Your Last Battlefield - The Enterprise exceeds Warp 10
The Changeling - The Enterprise goes to Warp 11, Weapons travel toward them at Warp 15
By Any Other Name - The Enterprise travels at Warp 11 with no ill affects
In TNG, the scale was recalibrated to make Warp 10 essentially unreachable. Speeds in excess of (approximately) 3000 times the speed of light are described as "transwarp speeds".