Learn Swedish: Det nya instrumentet låter fantastiskt idag. - The new instrument sounds fantastic today.

It describes instrumentet, which is a neuter (ett) noun. When using verbs of impression like låter (sounds) or verkar (seems), the adjective that follows must agree with the subject's gender, exactly as it would if the verb were är (is).

Why is there no som after låter?

You only use låter som (sounds like) when comparing the subject to a noun or a full clause, such as Det låter som en fågel (It sounds like a bird). Because fantastiskt is an adjective, you drop the som and just use låter directly.

If instrument is an ett word, why is the adjective nya instead of nytt?

Because the phrase is in the definite form (the new instrument). In a definite noun phrase starting with det, den, or de, the adjective almost always takes the -a ending, ignoring the noun's gender. If it were indefinite (a new instrument), it would be ett nytt instrument.

Doesn't låter also mean 'lets' or 'allows'?

Yes, the verb låta has two entirely different meanings. It can mean 'to sound' (as it does in this sentence) or 'to let/allow' (as in låt mig vara, meaning 'let me be'). The context of the sentence will usually make it obvious which meaning is intended.