So, here's the situation: A barbarian with beast totem pounces on a caster with frightful aspect, which states:
You become a larger, awful version of yourself. You grow to size Large, and take on features that horrify your enemies. You gain the following abilities: a +6 size bonus to Strength, a +4 size bonus to Constitution, a +6 natural armor bonus, DR 10/magic, and spell resistance equal to 10 + half your caster level. You also emit an aura that emanates 30 feet from you. Enemy creatures within the aura are shaken. Each time a creature shaken by this aura hits you with a melee attack, that creature becomes frightened for 1d4 rounds, though at the end of that duration it is no longer affected by this aura. The aura’s effect is a fear and mind-affecting effect.
So after the barbarian makes his first melee attack on the caster, he becomes frightened. The frightened condition states:
A frightened creature flees from the source of its fear as best it can. If unable to flee, it may fight. A frightened creature takes a –2 penalty on all attack rolls, saving throws, skill checks, and ability checks. A frightened creature can use special abilities, including spells, to flee; indeed, the creature must use such means if they are the only way to escape.
Frightened is like shaken, except that the creature must flee if possible. Panicked is a more extreme state of fear.
So does the barbarian get to make the rest of his attacks? He can't flee because he's already moved this round, but otherwise nothing else is preventing him from fleeing (say for the purpose of this scenario, there's nothing between him and the exit behind him.)
He can't flee at this moment, so may finish his attacks. This usually comes up when someone chases and corners a frightened target.