Learn Swedish: Mannen spelar instrumentet mycket skickligt. - The man plays the instrument very skillfully.

Because it is acting as an adverb, describing how the man plays. In Swedish, you generally form an adverb by adding -t to the base adjective. So the adjective skicklig (skillful) becomes the adverb skickligt (skillfully).

Doesn't adding -t to a describing word usually mean it goes with an ett noun? How can I tell it's an adverb here?

You are right that neuter (ett) nouns also force adjectives to end in -t. You can tell skickligt is an adverb here because it modifies the verb spelar (plays) rather than describing the instrument. It tells us how the action of playing is being done.

Why does instrumentet end in -et?

The suffix -et makes the noun definite, meaning "the instrument". Because instrument is an ett-word, its definite ending is -et instead of the -en ending you see on mannen (the man).

Does mycket always mean "very"?

Not always! When it comes right before an adjective or an adverb like skickligt, it translates to "very". But in other contexts, it usually means "much" or "a lot" (for example, tack så mycket means "thanks so much").