There is a common misconception about Stoics that they're nothing more than a bunch of apathetic and lofty people. Maybe it has something to do with our modern day definition of the word “stoic”, which means someone who is aloof and uncaring about what happens around them. This is of course wrong, but I can see where people might have gotten the idea.
The very basics of Stoic philosophy states that there isn't much in our control. It's easier to say what is in on control: our thoughts, opinions, and desires. Anything outside of that belongs to what we can't control. These things that we can't control the Stoics call “indifferents”.
So, doesn't indifferent basically mean uncaring? If anything we can't control is an indifferent, does that mean we shouldn't care about it?
Well, no. What the Stoic means when they speak of indifferents is that the object or event at hand have no bearing on living a good life. To the Stoics, Nature has given us reason, and with reason we can obtain virtue, and with virtue a good life (and, at least according to the later Roman Stoics, tranquility).
Does that mean we shouldn't worry about indifferents? Again, no. The Stoics saw a difference in indifferents: preferred and undesired. Even though a cave would shelter us, we'd prefer a house. Catching a cold didn't mean we couldn't work towards a good life, but that doesn't mean we'd go looking for illness. It's all in how we judge these things. If you think money buys happiness, you'll never have enough money. If you think you create your own happiness, you'll always be rich. Think dying is the greatest evil, you'll be too worried to live. Think dying is no thing at all, you'll have all the time to live. And so on and so on.
One way to think of indifferents is that they are like tools. We'd use a screwdriver to place screws, but we'd be silly to think it could hammer in nails or paint. We'd also be foolish to think the more screwdrivers, the better, simply because we needed one in the first place. And we wouldn't insist on using a screwdriver when only a hammer is needed. Sure, some people might get overly attached to one brand or another (another indifferent), but in the end, it's what works for you in that moment that's most important. And none of it would work without proper use of reason (like that whole “trying to paint with the screwdriver” thing. Reason prevents that sort of silliness).
Of course, the analogy falls a little flat when it comes to situations and people, both of which also counts as indifferents. We don't like to think of people as tools, which thankfully Stoicism doesn't promote, and we do often feel like we have at least some control over most situations. So how can it be these things are indifferents?
Remember, anything outside of our control is considered an indifferent. People, no matter how much we'd like to think about it, are always outside of our control. We can't help what they do, say, think, and so on. Insults, flattery, being cutoff in traffic – all these things don't bother the Stoic sage or inflate their ego. Now, we mere mortals aren't going to be prefect, of course, but the thing to remember is that most people aren't out to get us and, even if they were, we don't have any control over them. All we can control is what we think of other people and what they say and do. We can either become upset with them and ruin our day or think nothing of it and go on our day with little to no disturbance to our emotional state.
Okay, so maybe that makes some sense, but what about events in our life? Don't we have control over, say, getting to work on time? Sure, maybe we can't account for traffic and bad driving, but we do control some things. Well, not really. See, we are in control of our intention – our desire – to get to work on time, but that's it. Maybe we don't hear the alarm, or wake up and we find ourselves bound and gagged (my commiserations to your bad start, by the way). Even if we don't have any problems and do get to work on time, none of that was in your power. Once again, only your intention of getting to work was in your power.
The thing that needs to be remembered is that indifferent in the Stoic sense isn't one of apathy. The only thing it means is that they aren't either good or bad. Most of the time, it's how we see and think about the indifferents is what make them good or bad. So if it seems if a Stoic is unaffected by insults and maybe a little too uncaring about a underlings flattery, or if a flat tire doesn't disturb them, odds are it's because they decided it wasn't going to help them to live a good life to be concerned about it. But apathy it isn't.