Cicadas are one of the symbols for the Japanese summer.

Today, I felt that summer had arrived when I heard athe chirp of a cicada.

Alternative:
When I heard the chirp of a cicada today, it felt like summer had arrived.
(This version puts a bigger emphasis on the reason)

Another one is the evening cicada; they chirp in the evening as their name shows.

You can also use the word "imply" here, as in:
"Another one is the evening cicada; they chirp in the evening as their name implies."
Which I think is a bit more accurate.

The sound reminds me of my childhood, for example,like when I was getting back home after I played with friends until sunset.

Your sentence is grammatically correct, but "for example" sounds a little stiff in this context. "Like" makes the sentence flow a bit more nicely.