Learn Russian: Оба участника достигают высокого уровня мастерства. - Both participants achieve a high level of mastery.

The word оба (both) behaves grammatically just like the number два (two). When it is in the nominative case and describes a masculine or neuter noun, it forces that noun into the genitive singular. Since участник (participant) is masculine, it becomes участника.

Why is the object of the sentence (высокого уровня) in the genitive case instead of the accusative?

In Russian, certain verbs demand the genitive case for their direct object. The verb достигать (to achieve / to reach) is one of these verbs. Because of this rule, the phrase высокий уровень must change to высокого уровня.

There are two genitive parts in a row: высокого уровня and мастерства. Why?

This is a "chain" of genitives, which is very common in Russian. The first part, высокого уровня, is in the genitive because it is the object of the verb достигают. The second part, мастерства (from the neuter noun мастерство), is in the genitive to show a relation to the noun before it, translating as "of mastery".

Could we use the perfective verb "достигнут" here instead?

If you used the perfective достигнут, it would change the tense of the sentence to the future: "Both participants will achieve..." By using the imperfective достигают, the sentence describes a general fact, a present-tense state, or a repeated process.