What's the aim of avoiding attachment? I've recently started to research Buddhism, and one of the first area's I've been looking into is that of attachment. I found this quote -

"Seeing through the delusion of separation means we no longer give "external things" the power to make us miserable."

So does this mean we are supposed to feel emotionless to everything around us, only relying on ourselves for our emotions?
This seems to be a rather common misunderstanding, that Buddhism is a doctrine of emotionlessness or aloofness. Instead, (Mahayana) Buddhism is a doctrine of emotional maturity, emotional intelligence, mental agility, and robust practical excellence.

In (Mahayana) Buddhism we don't avoid attachments-as-connections, we still form relationships with people, make commitments, pursue long-term goals -- but we learn to let go quickly when situation changes, this saves us tremendous amounts of energy. That's why in English we call them "attachments", not "connections". Connections are alright.

Attachment is a much more broad notion than mere fondness of something good. It could be attachment to self-image, attachment to fairness, attachment to skills and intelligence, attachment to higher values, attachment to security, attachment to a certain perspective and so on. Every time we let go of attachment we let go of a button that "external" world may push to trigger an automatic reaction in us.

This is what that quote says, that when we ("I") consider ourselves a separate entity, we think we are entitled to our pathologies and neuroses. But when we understand that this is an ocean of information riding on top of various media, including the media of the primate brains, we can take ownership of our reactions, and choose not to get pushed around by the fluctuations of the signal. 

This does not mean we are frozen cold -- this means we can be positive, energetic, and happy all the time, because we now operate our own electrical generator that works both low tide and high tide, which makes us pretty independent and free.

Another metaphor I can offer is that of a sailboat -- once we learn to balance away the environmental influences, we are free to choose our course regardless of wind. It does not mean we lower our sails and stay in place, which is a metaphor for avoiding relationships altogether, and it does not mean aberrating randomly, which is a metaphor for avoiding long-time commitments, it just means we are not attached to one wind and one current, especially after it's gone.

Plus, "seeing through the delusion of separation" we no longer consider ourselves the center of the world. Once we stop obsessing about our own grief, it magically subsides, our eyes open, and we notice others stuck in all kinds of horror worlds. This is when it feels natural to go and help them out, but of course we must learn to float our own boat first.