놓다
1. lay, put, place
2. release, let go
3. build, construct, install
~고 = a conjunctive like 'and'
..I know you did not ask for translations but I hope the following will help in your understanding of 놓다 and 고
This sentence "내 맘을 찢어놓고" is essentially unfinished, grammatically speaking, but sometimes Koreans do just leave the completion of the thought to the listener. A literal translation would be something like "my heart ripped placed and" which is an extremely awkward translation of the idea 'You place my heart in a state of being torn to shreds and...' -looking at it again: You ripped up my heart and...' 'You/he/she ripped my heart out and...'
In addition to being an unfinished thought, the subject is also left unstated, something very common in Korean speech when it is assumed to be understood by the listener. In this case probably 'you/he/she' because it is very unlikely that the speaker has placed his own heart in such a state.
This second sentence "굳은 가슴 적셔 놓고 떠나가네요 비를 타고." is really difficult to translate into English because there are certain poetic nuances that just don't translate directly across. For example "굳은 가슴" could be translated 'stubborn heart' or 'hardened chest' or 'determined breast' and "비를 타고" is another expression for which there is no directly corresponding nuance, we could possibly say 'riding the rain' or simply 'with the rain' or even 'braving the rain' but they all seem to fall short of the feeling I get when I read the Korean. There is also a deliberate grammatical aberration by placing 비를 타고 as the final clause that gives a more poetic feel to the sentence.
Just as a very rough translation I would offer the following:
(first let me rearrange the sentence a little)
굳은 가슴 적셔 놓고 비를 타고 떠나가네요.
(here comes the funny part)
Humph! You depart (from me) in the rain with your stubborn chest soaked to the bone.
*Your determined breast is soaking wet as you depart, braving the rain.
(maybe someone can come along and embarrass me about my really poor translation)
hope that helps
George -- your fellow student