"Context" is merely the information that surrounds an idea or piece of evidence and helps a reader figure out how/why that idea or evidence is being used. In other words, "contextualizing" your writing means explaining the situation so that a reader can fully make sense of your specific examples.
Contextualizing Your Argumentative Topic
In general, people can't be persuaded about things that aren't immediately significant to them. If you want people to understand WHY they should be persuaded by your position, then, you generally need to make sure they understand the CONTEXT of your argument, by explaining why they should view the topic as relevant to their own lives.
If you write a paper arguing that electroshock therapy should be discontinued, for example, your reader might wonder why you're bothering to write a paper about this topic. But if you explain the current situation regarding electroshock therapy, you can help them understand that the topic is relevant. Perhaps new research has established an entire set of negative side effects that were previously unknown, but many therapists are unaware of that new research. Perhaps new legislation is making it much easier for relatively untrained people to offer electroshock services. Perhaps you never realized how common electroshock theraphy was until a doctor convinced your sister to try it as a treatment, and you feel that she's been completely changed by the experience. Perhaps the military is suddenly dumping tons of money into research into electroshock therapy, hoping to use it as a way to curb the tremendous rise in suicides of soldiers returning from combat in Iraq.
Often, there are LOTS of reasons why your audience might be encouraged to take an interest in the topic you're writing about. Once you've captured their interest by contextualizing the issue, they'll be much more open to rethinking their ideas about that topic.
Contextualizing Your Evidence
When you use evidence from sources that are unfamiliar to your readers, you need to make sure that you give them enough context for them to understand HOW and WHY you're using the information.
Imagine that a writer wanted to compare modern cancer treatment to Babe Ruth's fabled "called home run" in 1932. Perhaps you're really familiar with this historic event, but chances are good that your reader may not know the event. So, in order for your comparison to be effective, you need to fully contextualize the evidence so that your reader can understand HOW cancer treatment and a babeball game are similar.
In the past, cancer treatement was a "hit or miss" process. Doctors injected drugs into a patient and hoped that the drugs would kill lots of cancer cells and not too many good cells. But modern treatments are like Babe Ruth hitting his 1932 homerun. Research has really helped doctors kill fewer good cells, so treatments are much more effective.
The problem with this example is that the comparison isn't fully explained. For readers who aren't familiar with your evidence, the analogy is completely useless. Additionally, the analysis that follows the evidence doesn't relate clearly to the evidence. At present, a reader would probably have no clue how the evidence relates to the point being made in this paragraph.
The following revised paragraph attempts to address these problems by more fully contextualizing the evidence.
While cancer treament used to be a "hit or miss" process, new research provides doctors with methods that much more effectively target organs infected with cancer. In some ways, new treatments could be compared to Babe Ruth's 1932 home run. In a World Series game against the Cubs, Ruth stepped up to the plate, pointed his bat at deep center field, and then subsequently hit a deep blast directly to the location he'd just indicated (citation here). In contrast to other hitters, who merely hoped to hit a fair ball somewhere in the ballpark, Ruth clearly had much higher expectations for his accuracy. In a similar way, today's doctors expect extremely high accuracy from their cancer treatments. While previous treatments just tried to place drugs or radiation "in the ballpark" of the infected area, new treatments are able to focus in on very specific zones.
In this revised example, notice that the writing fully contextualized the evidence, then follows up by making comparisons that clearly show the relationship between the evidence and the topic of the paragraph. Good contextualization takes time, so don't be concerned if your paragraphs become significantly longer--that's probably a sign that you're adequately contextualizing and analyzing your evidence!