I sympathize with the anarchist tradition as well; however, I couldn't wrap my head around anarchism until I started learning about the things that are going on today within the anarchist movement. haymarket summarizes anarchism as 'political philosophy' but his description of anarchism remains utopian. in other words, anarchism is held up as a utopian system that we need to 'get to'. Talking about anarchism this way makes it hard to conceptualize what anarchism means in the present.
When I speak with people about anarchism, the most common reply is that anarchists are 'naive' or 'idealistic'. people assume that advocating anarchism requires assuming that 'all humans are good' or that 'if we got rid of all the laws, everything would be great'. These assumptions stem from treating anarchism as a utopian political philosophy that is 'somewhere in the future' rather than right now.
However, the most interesting part of anarchism (to me) is its strength as a philosophy for the present. When we start thinking about anarchism in the present, it opens up a whole new set of possibilities that are often completely invisible to the mainstream. These possibilities are hard to think about because they don't fit into what most people traditionally think of as 'politics'.
For example, as I write this, all of the subheadings under politics are about policy and governmental politics: elections, energy policy, foreign policy, healthcare, and Iraq. For a lot of people, these are the limits of politics; we get to talk about what 'the politicians' or 'the policymakers' are doing, but we don't do much ourselves, other than argue about policy, and vote every few years.
Anarchism helps open a whole new field of 'politics'. A lot of people who don't identify or sympathize with anarchism actually practice forms of politics that are 'anarchistic', meaning that they are based on a lot of the same principles that are central to anarchism. For example, a lot of people (not just anarchists) practice 'direct action'. This means getting out there and being political yourself, or within your own community. Direct action describes political acts that don't rely on government or other large-scale institutions.
If you're against the barrage of corporate advertising that saturates all of our cities, direct action might mean spraypainting a corporate billboard, rather than lobbying the city for new laws. If you're looking to take a stand against pollution, global warming, and huge SUVs, direct action might mean joining in a Critical Mass ride (google it) rather than lobbying government for more bike lanes or asking them to sign onto Kyoto.
Direct action doesn't have to be illegal or 'radical'. Community gardens are a great example. These might not seem to be 'anarchistic' at all... what's political about gardening? When we start to think about it, community gardens can be deeply subversive because they challenge the monopoly of huge corporate agribusiness and factory farms. For people who oppose corporate farming (maybe because of its pollution, treatment of animals, or because it ruins the livelihood of small-scale farmers) community gardening provides an alternative that lets us get (at least some of) our food elsewhere.
The idea of providing alternatives is important as a concept of anarchism, and as a way of doing politics in general. Direct actions like these help to show that other worlds are possible. In other words, they help to show that the basic ideas behind anarchism (more freedom, more autonomy, more creativity) aren't so naive or idealistic after all. This is political in itself because it presents alternatives to neoliberalism, corporate control, and reliance on government for help with our problems.
A final note: widening the concept of 'politics' doesn't necessarily mean throwing out traditional ways of doing politics. In other words, just because we recognize that gardening can be political and that it can challenge corporate farms doesn't mean that we should stop challenging corporations and governments in other ways. In other words, sympathizing with anarchism doesn't necessarily mean asking for governments to dismantle welfare and social services. We need to recognize that this would play into the hands of corporations because people depending on these services would be forced to pay for private services instead. However, we can chip away at this dependence by creating alternative ways for people and communities to sustain themselves. This won't happen all at once, but the important (and inspiring) thing about direct action is that it's happening somewhere right now!