What does 懐かしい Natsukashii mean?

Natsukashii is what a person says on encountering something from their past. It is an adjective, and it attaches to the object rather than to the person, which is the first of two differences from English nostalgic.

Where the word points

English puts the state in the person: I feel nostalgic. The Japanese construction puts the quality in the thing: this song is natsukashii, this street is natsukashii. The object is described as having the property of returning the past.

English can do this occasionally, calling a photograph evocative, but the standard construction runs the other way, and the difference changes what a sentence is about. One reports an internal state; the other makes a claim about the thing in front of you.

The second difference is the temperature

English nostalgia has sadness in it. The word arrived through a term for homesickness treated as an affliction, and the modern sense keeps a note of loss: something is gone, and the pleasure is complicated by that.

The Japanese word is warm and largely uncomplicated. It is closest to the English exclamation on finding an old toy, where the reaction is delight rather than grief. The past being past is not the point of the sentence.

Why the subtitle usually misses

Because there is no English adjective for it. Nostalgic is the only candidate and it drags in both problems at once, relocating the quality to the speaker and darkening the mood.

A more accurate rendering is often an exclamation with no adjective at all: oh, this takes me back. That is what the Japanese line does socially, and it is why subtitlers who choose it produce a better result than the dictionary word.

What the difference implies, carefully

Not that one culture is happier about the past. Japanese has plenty of vocabulary for loss and regret, including a well-known aesthetic term for the sadness of transience.

What it has in addition is a common everyday adjective for the pleasant half of remembering, unmixed with mourning. English fused those into one word, which means an English speaker reaching for the pleasant half arrives holding the sad half too.