It doesn't "mean" anything, any more than the present participle of a verb in English (the "-ing") form "means" anything. Like the present participle, it has a number of different grammatical uses. The main thing it does is to connect the verb to another verb or clause.
本屋へ行って、本を買った。Honya e itte, hon wo katta. (I) went to the bookstore and bought a book.
The "-te" form of "iku", which is "itte", connects the first and second clauses, as the word "and" does in English. Since the final verb is in the past tense, the tense of the entire sentence is past tense.
今食べている。Ima tabete iru. I am eating (now).
The "-te" form of "taberu", which is "tabete", connects the verb "taberu" to the state of being verb "iru". The combination "~te iru" expresses being in a state of doing the verb ~ over some period of time. Here the word "ima" says the period of time is the present time; the combination "ima ~te iru" is like the present progressive tense "is/am/are ~ing" in English. However, with a different time specification, "~te iru" can also be used to express what we express with the present perfect tense in English:
最近勉強している。Saikin benkyou shite iru. (I) have been studying recently.
食べてください。Tabete kudasai. Please eat.
Here the connection is to the imperative form of the polite verb "kudasaru", meaning "give (to me)" or "do (for me)".
This is just meant as an introduction to prevent you from latching onto any single "meaning" for the "-te" form. Here is a place you can learn more about its uses:
http://www.guidetojapanese.org/learn/grammar/compound
http://www.guidetojapanese.org/learn/grammar/teformDecember 19, 2015 at 10:16am