How can creatures with indestructible end up in the graveyard?
Indestructible prevents the permanent from being destroyed.  This usually is caused by damage exceeding a creature's toughness, or an effect that explicitly says that it will destroy a permanent.
The main ways that a creature with indestructible will die (being put to the graveyard from the battlefield) will be:

Having its toughness reduced to zero via -X/-X effects. (Tragic Slip, Black Sun&#39;s Zenith or -1/-1 counters, including damage from a source with infect)
Being sacrificed. (Innocent Blood)
Losing indestructible and being destroyed (Hour of Devastation, Turn to Frog + destruction effect)

Other ways to deal with indestructible permanents that do not cause them to &quot;die&quot; include:

Exiling. (Swords to Plowshares)
Bounce. (Unsummon)
Interacting with them on the stack. (Counterspell)
Discard. (Thoughtseize)
Removal from the deck. (Memoricide)
Taking control of them. (In Bolas&#39;s Clutches)
Move them to their owner's library. (Condemn)