успокоившись is the past GERUND = having calmed down / having relaxed.

успокоившийся is the past active PARTICIPLE = he who had calmed down / he who had relaxed.

You need to check the difference between Participles and Gerunds.

Participles are verbal adjectives, that is 'adjectives formed from verbs'. They therefore agree in case, number and gender with the noun they describe. Note they are commonly used by themselves sort of as nouns. They replace a clause with который:
Девочка, которая читает - The girl who reads / The reading girl / The girl who is reading.
Читающая девочка - The girl who reads / The reading girl / The girl who is reading.

Успокоившись, он стал говорить тише.
Having calmed down, he began to talk more quitely

Успокоившийся ребенок мирно играл.
The "having calmed down" child played peacefully.
The calmed down child played peacefully.
The child who had calmed down played peacefully.