Learn Russian: Чем выше уровень мастерства, тем быстрее мы достигаем цели. - The higher the level of mastery, the faster we achieve the goal.

It creates the "The [comparative] ..., the [comparative] ..." pattern. Russian doesn't use a word for "the" here. Instead, it literally means "By what (degree) higher..., by that (degree) faster...". In this sentence, it pairs with the comparatives выше (higher) and быстрее (faster).

Why doesn't the first part of the sentence have a verb?

In the present tense, the verb "to be" is typically omitted in Russian. The sentence literally translates to "The higher the level of mastery [is]...". If you wanted to say this in the past tense, you would need to insert the verb: Чем выше был уровень....

Why is the word for "goal" (цель) written as цели here? Shouldn't a direct object be in the Accusative case?

Normally, yes, but the verb достигать (to achieve or reach) is an exception. It strictly requires its object to be in the Genitive case. Since цель is a feminine noun ending in a soft sign, its Genitive form is цели.

Are выше and быстрее both adverbs in this sentence?

Not quite! выше (higher) is a comparative adjective describing the noun уровень (level). быстрее (faster) is a comparative adverb describing the verb достигаем (we achieve). In Russian, short-form comparative adjectives and comparative adverbs are usually identical.