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この - this (something or someone close to the speaker, including the speaker, or ideas expressed by the speaker) (In this case, the meaning is more or less "erm," as the character is about to start his report).
ここ - 1: here (place physically close to the speaker, place pointed by the speaker while explaining); this place (within this context).
の - indicates possessive and nominalizes verbs and adjectives
ホーム - platform (within this context)
で - indicates location of action; at; in (within this context)
二人 - ふたり - two persons; two people; pair; couple (2人 has the same meaning as 二人)
高校生 - こうこうせい - senior high school student
が - indicates sentence subject (within this context)
線路 - せんろ - line; track; roadbed
に - Indicates such things as location of person or thing (within this context).
落ちる - おちる - to fall down; to drop; to fall (落ちた here is the plain past tense version of 落ちる - when placed before a noun, this form can be used to indicate the subject of the action of the verb. For example here, 線路に落ちた人間 means, a human being that fell onto the track. As another example, 渋谷で食べたステーキは美味しかったです [the steak eaten in Shibuya was delicious] - 食べたステーキ meaning the steak that was eaten [食べる meaning to eat])
人間 - にんげん - human being
を - indicates direct object of action (within this context)
助ける - たすける - to help; to save; to rescue; to give relief to (助けた here is again in the plain past tense form, but when followed by 後, this expresses the idea that the following actions described took place afterwards. For example, 渋谷でステーキを食べた後、家に帰りました [After I ate a steak in Shibuya, I returned home] - ステーキを食べた後...xyz, meaning after the steak was eaten xyz occurred).
後 - あと - after; later
忽然 - こつぜん - sudden; unexpected (this when used as an adverb requires the 'と' particle, making the meaning into suddenly or unexpectedly)
姿 - すがた - figure; form; shape; appearance (within this context) (this phrasing doesn't translate very comfortably into English, 姿を消す means literally to erase or delete their appearance, but in translation saying disappeared would probably be a close approximation)
消す - けす - to erase; to delete; to cross out; to extinguish (when adding しまう to the end of the て form of a verb, within this kind of context this creates the nuance of a negative result, so the characters regrettably or unfortunately vanished. For further uses of てしまう, and for further examples of this usage, please see http://ww8.tiki.ne.jp/~tmath/language/jpverbs/lesson68.htm)
のです - this highlights the subject or situation at hand and is also used to seek confirmation of current circumstances. (です is the polite copula in Japanese and roughly means "is, be, are, am". It is very common in Japanese and usually put at the end of a statement - for a further description please see http://ww8.tiki.ne.jp/~tmath/language/jpverbs/lesson37.htm)
初冬 - はつふゆ - early winter
ミステリー - mystery
地下鉄 - ちかてつ - underground train; subway
ホームレス - homeless, homeless person
救う - すくう - to rescue from; to help out of; to save (救った is in the plain past tense)
消える - きえる - to go out; to vanish; to disappear (消えた is in the plain past tense)
On this platform here two senior high school students, after saving a human being who had fallen onto the tracks, unfortunately disappeared suddenly.
Early Winter Mystery
Two senior high school students who saved a homeless person in an underground train station have disappeared?!