In an LC oscillator, if you charge up the capacitor (DC source) and disconnect the power, the capacitor will produce a current that increases through the inductor. THe current increases, but the rate at which it increases slows down as the capacitor discharges.
When the capacitor is discharged at 0V, the inductor now has peak current going and it can't just immediately drop to zero because that cna't happen in an inductor. The inductor now has stored energy, but the capacitor is 0V and is in parallel with the inductor. That's the same as an AC short-circuit across the inductor. So the inductor releases it's energy by forcing current through the capacitor. Since current can't change immediately in an inductor, the current level begins at where it left off at peak levels (the capacitor also most like a short-circuit since it is 0V across the inductor so the most current would flow anyways). THe inductor current charges up the capacitor voltage, starting at peak current and decaying at an ever slow rate approaching asymptotically to zero.
When the capacitor is charge d up and the current is zero, the whole process begins again. THe circuit oscillates as energy jumps between capacitor and inductor. One half of the period: capacitor gives energy to inductor. Other half of the period: inductor gives energy back to the capacitor.
Now...imagine that instead of a DC source that charges up the capacitor and gets disconnected (otherwise capacitor voltage would never drop and the inductor would eventually become a short-circuit across the DC source), you have an AC source that stays connected.