When you flashback a spell you will exile it afterward no matter what the result is. For example Remand would normally return a countered spell to a players hand, but if the spell was flashed back it would be exiled.
It gets exiled if it is countered. Casting a spell for its flashback cost creates a replacement effect that makes the card go to the exile zone instead of any other zone it could go to if it would leave the stack for any reason. (This even applies to corner cases like Venser, Shaper Savant where a spell is removed from the stack without either resolving or explicitly being countered.)
The phrasing "Then exile it" is reminder text, but you need to look at the official rules definition of the ability to have the details about how it works.
702.33a
Flashback appears on some instants and sorceries. It represents two static abilities: one that functions while the card is in a player’s graveyard and another that functions while the card is on the stack. “Flashback [cost]” means “You may cast this card from your graveyard by paying [cost] rather than paying its mana cost” and “If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack.” Casting a spell using its flashback ability follows the rules for paying alternative costs in rules 601.2b and 601.2e–g.
The reminder text of flashback isn't technically correct in order to be more concise.
The comprehensive rules says this about flashback:
702.33a Flashback appears on some instants and sorceries. It represents two static abilities: one that functions while the card is in a player's graveyard and the other that functions while the card is on the stack. "Flashback [cost]" means "You may cast this card from your graveyard by paying [cost] rather than paying its mana cost" and "If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack." Casting a spell using its flashback ability follows the rules for paying alternative costs in rules 601.2b and 601.2e-g.
So countering a spell cast with flashback still results in it being exiled.
The wording is what confuses me here. The wording is "You may cast this card from the graveyard for its flashback cost. Then Exile it."
Is exiling it part of the cost of using flashback, or the result of using flashback?
The phrasing "Then exile it" is reminder text, but you need to look at the official rules definition of the ability to have the details about how it works.
The comprehensive rules says this about flashback:
702.33a Flashback appears on some instants and sorceries. It represents two static abilities: one that functions while the card is in a player's graveyard and the other that functions while the card is on the stack. "Flashback [cost]" means "You may cast this card from your graveyard by paying [cost] rather than paying its mana cost" and "If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack." Casting a spell using its flashback ability follows the rules for paying alternative costs in rules 601.2b and 601.2e-g.
So countering a spell cast with flashback still results in it being exiled.
I thought that may be the case. The wording on the card just made it seem like it may be the result of flashback and not the cost of flashback.