Learn Russian: Встречающие ждут поезд на платформе. - The greeters are waiting for the train on the platform.

It is a substantivized participle, which means it is a participle (a verbal adjective) acting as a noun. It comes from the verb встречать (to meet, to greet). While it literally translates to 'those who are meeting', it is the standard noun used at stations and airports for 'greeters' or 'people picking someone up'.

How exactly is the form встречающие built from the verb встречать?

Present active participles are built from the 'they' (third-person plural) form of the present tense verb. 'They meet' is встречают. You drop the final -т and add the participle ending -щий (or -щие for plural). Since it functions as a noun here, it declines just like a standard plural adjective.

I learned that the verb ждать takes the genitive case. Why is поезд in the accusative case here?

You generally use the genitive with ждать when waiting for something abstract or an indefinite, random object (like waiting for a bus in general). However, when you are waiting for a specific, concrete object or a specific person, you use the accusative case. Because the greeters are waiting for a specific scheduled train, поезд takes the accusative (which looks exactly like the nominative for inanimate masculine nouns).

Is there a similar substantivized participle for the people who are dropping someone off at the station?