Saying that the -는 ending creates an adjective can be a bit confusing. Technically, it creates a participle, which is the form of the verb which acts like an adjective, that is, by modifying a following noun.
These are primarily used in Korean to create relative clauses, for example:
멀리 보는 사람들이 성공할 수 있다
People who see far ahead can be successful
Here, the relative clause in English "who see far" comes after the noun it modifies "people", but in Korean, it's always the opposite: the participle "보는" comes right before the noun "사람".
Looking at your sentence:
음절의 끝소리가 되는 자음
We can see that the main noun is 자음 (consonant), the participle is 되는 (that becomes), which has the subject 음절의 끝소리가 (the final sound of the syllable), which put together is:
consonants that (can) become the final sound of the syllable.
Looking at the complete sentence, it says:
한국어의 모든 자음은, All of the consonants in Korean
받침으로 사용할 수 있지만, can be used in the bottom of the (written) syllable but
음절의 끝소리가 되는 자음은 ‘ㄱ, ㄴ, ㄷ, ㄹ, ㅁ, ㅂ, ㅇ’의 7가지입니다 the consonants that can become the final (spoken) sound of the syllable are just 7 (ㄱㄴㄷㄹㅁㅂㅇ).
These relative clauses formed by participles are a very important part of the Korean grammar. They are very flexible, so that the participle can be preceded by a subject (as in your example) or an object. See the difference:
강아지가 먹는 고기 (meat that the dog eats)
고기를 먹는 강아지 (the dog that eats meat)
It's important to also know the past participle (ending in -ㄴ) and future participle (ending -ㄹ):
강아지가 먹은 고기 (the meat eaten by the dog)
가아지가 먹을 고기 (the meat the dog will eat).