This is the 332nd most frequent Chinese word.
随着
"Along with; as; following."
On this context, '随着' in '随着时代的发展' introduces an action or event that happens accompanying or as a result of another event or condition.
随着时代的发展，人们的生活质量有所提高。
With the development of the era, people's quality of life has improved.
Here, '随着' in '随着年龄的增长' denotes a process or change occurring concurrently with another process or event.
随着年龄的增长，他学到了更多的智慧。
As his age increases, he has gained more wisdom.
In this sentence, '随着' in '随着音乐的节奏' indicates following or in compliance with the rhythm of the music.