Why are there high frequency components from an LC oscillator
A BJT Colpitts oscillator (for example) relies on non-linearities for keeping the output signal from rising and rising until the BJT saturated (undesirable). This produces harmonics in the output waveform and, in your FFT, it looks like the 2nd harmonic is about 20 dB lower than the fundamental. This is pretty much what I would expect with a simple circuit and nothing really to worry about. The sine wave will be a little asymmetrically clipped but, for most target applications, this is of no consequence.


  where are the high-frequency parts coming from?


From the non-linearities needed in the circuit to maintain reasonable amplitude stability and avoiding BJT saturation.