What is the proper understanding of "taking refuge" in Buddhism?

Taking refuge is not about taking on a boss, far from it. The Buddha says many times in the Pali suttas that only you can put in the work to free yourself. Taking refuge can be seen in the same way as when someone may take refuge from a storm, looking for help and protection. The Triple Gem can be that refuge in the storm of life

As Andrei rightly pointed out in the other answer traditionally you take refuge by reciting the words "I go to the (buddha/dhamma/sangha) for refuge". This is often put together with the precepts for the ceremony to "become" a Buddhist. There is much more to it though, as seen in this wonderful treatise on going for refuge by Bhikkhu Bodhi:

http://www.accesstoinsight.org/lib/authors/bodhi/wheel282.html

But what is the going for refuge? At first glance it would seem to be the formal commitment to the Triple Gem expressed by reciting the formula of refuge, for it is this act which marks the embracing of the Buddha's teaching. Such an understanding, however, would be superficial. The treatises make it clear that the true going for refuge involves much more than the reciting of a pre-established formula. They indicate that beneath the verbal profession of taking refuge there runs concurrently another process that is essentially inward and spiritual. This other process is the mental commitment to the taking of refuge.

I will say that in my own experience I took the ceremony of refuge a few years before I had advanced in my practice and confidence of the triple gem to really mentally take refuge. The Refuge of the Triple Gem becomes powerful when you understand the reality of things enough to see how important the practice is and how full of truth the dhamma is.I view the triple gem as follows:

The Buddha: The Buddha is the "awesome dude" whom I often call "Sid". I take refuge in him because he was just a regular guy like myself who did an amazing thing, found a way out of samsara and endless suffering to show us how to make our lives better. He became awakened, and that is something I'd like to work on myself so I take refuge in him and all the awakened ones of past and future who are the embodiment of dhamma.

The Dhamma: Straight up truth, the reality of life, the way things actually are, not how we perceive or want them to be. Of all the triple gem, Dhamma is the most important, this is why the Buddha called what he taught " dhamma-vinaya" and said the Dhamma should be our master. How can you not take refuge in truth?

The Sangha: Specifically the Aryan(awakened) Sangha and the to a lesser extent the rightly practicing monastic sangha. These men and women are the keepers of dhamma. They have been making sure the Dhamma found by Sid 2600+ years ago survives today for us to practice. Rightly practicing monastics are the heart of the teaching and in supporting them you not only get a refuge, but you ensure the triple gem itself survives to provide refuge to all beings.

With confidence in your own practice you have confidence in the triple gem, which leads to more confidence in your own practice as you begin to prove right what the Buddha taught. At first it is just something you say because "thats what buddhists do" but it becomes so much more than that. Then there is no real point to taking refuge in anything else, because anything else you can take refuge in cannot bring you lasting happiness, it is a shoddy refuge at best, prone to fall apart in the weakest of storms, but when you protect the dhamma, and are protected by the dhamma, that is a refuge not even death can penetrate.

By taking refuge you are aspiring to build the qualities of the Buddha, Dhamma and Sangha in your self.

Before talking about Buddhist refuge, it is useful to recognize that all people everyday are taking refuge in various things. Let me explain.

Everyone wants to be happy and avoid suffering. People engage in various activities because they hope it will give them some sort of happiness and security. They go to universities to gain knowledge, they work in companies to earn a lot of money, they find partners and set up families or they spend most of their time pursuing some hobby. They do it because they believe this is what will give them happiness and this is something they can rely on. So in other words they take refuge in it.

All those things are great and can give lots of happiness indeed but the problem is that they are not permanent. One can easily lose a job or a partner and even if one has a seemingly perfect life, illness and death will inevitably come one day.

Gautama Buddha deeply understood impermanence - that everything conditioned would eventually fall apart so he persistently searched for something unconditioned, permanent and unbreakable, something that can give lasting happiness. And this unconditioned thing is Buddhahood - enlightened state of mind or one can say - true nature of our mind.

Taking refuge in Buddha means opening up to impermanent and perfect qualities of our mind. Buddhahood is our goal. The methods which will bring us there is the Dharma and friends, our helpers on the way, is the Sangha.

Taking refuge in Buddha is actually a great responsibility - we stop hoping that others will make us happy or thinking that the source of our suffering is current political situation or neighbours. We admit that the only lasting happiness is to understand the nature of our mind and it will depend only on us whether we succeed in it or not. Dharma and Sangha are there to assist us but it is entirely up to us whether we use it or not. After all, we are the bosses and everyday we are free to decide what we want to take refuge in.

Traditional refuge formula goes like this:

Buddhaṃ saraṇaṃ gacchāmi.
Dhammaṃ saraṇaṃ gacchāmi.
Saṅghaṃ saraṇaṃ gacchāmi.

To Buddha for refuge I go.
To Dhamma for refuge I go.
To Sangha for refuge I go.

According to dictionary, "sarana" means den or lair, but also more broadly, shelter or refuge. Apparently, at the time of Buddha, "sarana" meant being under political protection of someone. For example, when a peasant acknowledged becoming a subject of, say, king Bimbisara, he would say "Maharajam Bimbisaram saranam gachami" -- literally, "To great king Bimbisara for refuge I go". This would serve to indicate that A) the peasant accepts no other lord and won't pay taxes to anyone else and B) the peasant now expects the king to provide military protection from bandits and other neighboring lords.

The same way, when beginner Buddhist goes for refuge to Buddha, Dharma and Sangha, he or she A) promises to accept no other religious or philosophical authority as higher than Buddhism and B) expects to find protection from worldly troubles in Buddhist practice.

In Tibetan Buddhism, this idea of taking up a "provisional boss" (my play on your words, not theirs) is said to be a temporary measure employed for the duration of training, until one gets to the ultimate protection, the luminous mind of Bodhicitta.

The above is my own understanding, at the same time I really like explanation given by Chogyam Trungpa, which directly dismisses your dilemma about the notion of refuge contradicting Buddha's imperative of one being one's own island:

By taking refuge, in some sense we become homeless refugees. The point of becoming a refugee is to give up our attachment to basic security. We have to give up our sense of home ground, which is illusory anyway. We are suspended in a no-man's land in which the only thing to do is to relate with the teachings and with ourselves.

The refuge ceremony represents a final decision. The ceremony cuts the line that connects the ship to the anchor; it marks the beginning of an odyssey of loneliness. Still, it also includes the inspiration of the preceptor and the lineage. Acknowledging that the only real working basis is oneself and that there is no way around that, one takes refuge in the Buddha as an example, in the dharma as the path, and in the sangha as companionship.

-- basically, he defines taking refuge as becoming a (metaphorical) mendicant, which refers to letting go of one's stereotypes about oneself and reality, and the false sense of security they provide. As one thus becomes truly placeless (radically open-minded), the extent to which one takes comfort in Buddha is only as inspiration of predecessor having made the same path.

At that particular point, the energy, the power, and the blessing of basic sanity that has existed in the lineage for twenty-five hundred years, in an unbroken tradition and discipline from the time of Buddha, enters your system, and you finally become a full-fledged follower of buddhadharma. You are a living future buddha at that point.