Firstly, the circle of fifths links together related keys. Two keys that are adjacent on the circle differ only by one sharp or flat. In European classical music, there often occur key changes that follow the circle of fifths. For instance, a piece that starts in C major might modulate to G major. Key changes that follow the circle of fifths can be more or less easily "concealed" by the composer.
Besides modulation, harmonies that follow the circle of fifths have a strong sense of forward motion. The V-I cadence (dominant to tonic resolution) follows the circle, and so does the II-V-I that often appears in jazz and other genres: for instance, Dm G7 CM7.
The circle of fifths steps through all notes in the twelve tone system, and so it can be used as a pathway to reach any tonality. If you want the music to change from G to G#, you can do it abruptly, or you can get there in other ways, one of which is the roundabout trip through the circle of fifths, whereby you can make the changes so smooth, the listeners are hardy aware they have been taken from G to G#, in comparison to an abrupt change by a half step.
Re: finding out key
Finding out the key of a song doesn't have much to do with the circle of fifths. If the music exhibits tonality and is all in one key, then you can simply play along with the song by ear, until it becomes obvious which scale is being used, and what is the root note to which the music keeps returning. If the music uses some familiar melodic or harmonic devices, that of course helps, and some of those may be based on the circle of fifths. For instance if you recognize a II-V-I cadence being played, that may be a hint about the tonality of the music, or at least of that section of the music.
To find the key, or at least the starting key, of written music, you have to be able to read the key signature. This is linked to the circle of fifths, but you don't have to use the circle in any way. There are shortcuts to mapping a correctly written key signature to the key. When it comes to the flats, the first key (B flat) is F. The other ones are all given by the previous flat. The next flat to be added is E flat, and the key is then B flat: the previous flat. For the sharps, the rule is that each new sharp is the leading tone of the new key (leading tone being the half step below the tonic). So look for the last sharp in the key signature, and the half step above that is the key.
Recognizing key changes in the middle of music (heard or written) is tricky because you have to work out whether an outside note is part of an enduring key change, part of a brief harmonic device that borrows from another key, or some other alteration like chromatics or parallel modes.
Re: application to Indian Classical Music
The circle of fifths does not apply very well outside of the European music system, where it is a necessity due to the well developed system of modulation (and tuning compromises for its sake). Modulation is less important in Indian music, and detailed tuning is more important. It has a large number of scales which cannot be rendered properly on twelve-tone, equal-temperament instruments. A great resource about Indian music is Dr. Vidyadhar Oke, and his website 22shruti.com. Through numerous lectures, he reveals a tuning system based on 22 tones ("shrutis") which, he argues, contains the intervals which can represent not only the scales used in Indian music, but also the pure scales from other cultures (though obviously not equal temperament, with its frequency relationships based on irrational numbers).
Note that being able to return to the original note in twelve steps around the circle of fifths only works because some of those steps, or perhaps all of them, are not actually, pure fifths. A compact way to follow the cycle of fifths is to go up a fifth, and down a fourth and so on. If you do this six times, you should arrive at the original note, but an octave higher. If you follow pure fifths and fourths (frequency ratios of 3/2 and 4/3), then you end up at a note which is off pitch, sharp by about 1.36 percent. This discrepancy is the Pythagorean comma. Different historic tunings distribute this comma between the notes in various ways to conceal it. Equal temperament conceals the Pythagorean comma by making all twelve semitones exactly equal. The fifth becomes a leap of seven of these equal semitones, a frequency ratio of 2^(7/12) (two to the exponent 7/12), which works out to about 1.498: slightly flatter than 3/2.
Other historic tunings such as Well temperament take different approaches, which leads to fifths of different sizes being found around the musical keyboard, making the same harmonic devices sound different when transposed to different keys.