(SPOILER ALERT, so move on if you haven't or want to see it).
First, for small kids six and under, the movie is probably fine. Most of what I analyze here will go over their heads, and they will see a see nothing more than a cartoon with toys that come to life and go through some adventures - some a bit scary - but no worse for wear.
It is the older kids and pre-teens that are going to have difficulty interpreting this movie. And the thing they are going to ask questions about is the main "bad guy", Lots of Huggin' Bear.
BUT it is not really his fault that he is bad. Really. It was because he was abandoned. And his fear of abandonment makes him emotionally distant and unwilling to bond to others. And he views with contempt others who want to bond, because he believes they, too, will be abandoned in the end. So while he might come off as bad and mean, he is actually just trying to protect himself and others from being abandoned.
So he has that.
Plus he controls a little toy enclave for the benefit of himself and a few cronies who keep him in power. They work a system at a preschool where new or older toys are thrown to the mercy of the "toddlers" while he and a chosen few are played with nicely by older kids (and don't get broken). But it is not out of sadism that he is doing this, but the belief that all toys in the community will be permanently damaged if they were all in the room with the toddlers. So a few toys, especially the new ones, have to do their time in the small kids' room. Maybe some get moved out eventually. Most don't. But a few have to be sacrificed so that others (he and those he chooses) may continue to lead a more comfortable life. He'd fit in quite nicely in the upper echelons of most EU governments.
But he gets his just deserts at the end, being stuck on the end of the grill of a large garbage truck. The problem here is that this scene reminded me of the torturing of people in "Mad Max 2". Not a good way to go and not exactly kid appropriate, even if they don't know the reference.
Even without torture, the movie has a lot to digest for an 8 or 9 year old. The hardest part for them may be understanding how someone who looks and acts nice can really be mean. But that is actually a good lesson for a kid to learn.
An Adult Existential View
At a larger level there is the question of what sentient toys think should happen to their "lives" after their owner grows up. They know something will happen to them - attic, garage sale, trash, even a museum as seen in Toy Story 2 is a possibility, but they all believe these are bad ways to continue. They cling to the idea of a permanent state with the original "owner", while they watch the very same owner grow up and play less and less with them. Their best possibility to be handed to another kid to re-live their original experience, but these new kids will also grow up and leave them. So they cling to a reality which can never be in the long-term, and because of that they are always worried, searching for that perfect, permanent owner, never to be fulfilled.
These toys have not accepted their existential meaning, which is to exist for the pleasure of a child for only a short period of time, then knowing their usefulness and existence will end. So they need to embrace that time they have with the owner and enjoy it to the last, and not worry about non-existence later.