How LC oscillators work with a continous DC power supply
Oscillators that run continuously require something somewhere to have gain at least a bit over 1 at the oscillating frequency.  A L-C tank does not oscillate continuously on its own.  It can if connected to the right amplifier in the right way.

Think of any amplifier with a gain over 1 where the output is AC coupled back to the input.  There will always be some noise, so the output moves slightly, let's say it goes up for sake of example.  That drives the input up, which makes the output go more up, which makes the input go more up, etc.

Eventually the amplifier output hits a limit and can't go up anymore.  Now the input is no longer changing.  Remember that the input is AC coupled to the output.  A flat level on the output results in a 0 level on the input eventually.  With the output clipped no longer changing, the input decays towards 0.  At some point, the input isn't high enough to keep the output at maximum, so the output begins to fall.  That falling output makes the input even lower, and the whole process repeats, this time with the output at its minimum level.

That's the basics of how oscillators work.  The details are in getting them to produce a nice wave shape and predictable frequency.