First, you never regard any source as "true" or "false." Those really aren't very useful words for how one looks at sources. Instead, you have to consider the context of things; history is all about context, after all.

"Context," for historians, means several things. It means looking at how not just one source describes something, but how a whole body of them does. The views of one writer or witness don't actually teach us a whole lot, but if we have twenty--which will all be different--we can start to learn something from the commonalities and differences anong them. This leads to another important part of "context," the intentions of the writers. No one in the past is writing for our consumption (not many, anyway). They're writing for their time and place, whether it's a government report or a novel or journal entry. Understanding what the goals of people are helps us understand why they might write about things in particular ways. And it's always more complex that simply "bias," because EVERYONE is biased. There are NO objective accounts of anything, since everything was produced by someone.

So, the basic answer to your question is that a historian assesses sources by reading a lot of them, and reading them against one another, with the contemporary situation in mind. No source exists in a vacuum.

/r/agentdcf does a great job here. I would add that another way of looking at sources (primary or secondary) is "source criticism." This is perhaps another way of saying "context," the term /r/agentdcf uses to great effect. The term "source criticism" places a focus on understanding everything possible about the source - its author, its historical context, its political context, and it goes beyond context in a way, by suggesting that one must never let one's guard down when evaluating - or critiquing - a source. As /r/agentdcf points out, everyone is biased in some way.

As the Positivists of the early twentieth century took on the mantle of science as they looked at the past, they wrestled with the question of how one could deal with human subjectivity that worked its way into every aspect of history - from primary to secondary sources. How could one deal with all this subjectivity and still be a science - that is, how could historians know when they were dealing with the "truth"? It was easy to see that in many ways, the answer was, never, but that was not good enough for the Positivists. The French Annales School attacked this problem by backing up and looking at history at a different angle.

For example, if primary sources of the 1990s reveal a great deal about popular conspiracy theories, but sources of the 1890s do not, then the historian can argue that there has been a shift in the popular attitude of people in general (what the French called the "mentalités" - although see the essay in the link provided for a critique of this approach). What this says, then, is that there may be a debate over whether the USAF is covering up alien encounters or whether several people conspired to kill JFK - and these questions may never be settled to the satisfaction of everyone, but there can be no doubt that a significant number of people believed there were (and are) conspiracies in these (and a whole range of other) matters. And the growth of conspiracies to suit many different aspects of history points to a general attitude. What is the truth? - if we can't agree on that, we can agree that a lot of people in the Post Modern world believe that their governments are engaged in all sorts of conspiracies that withhold the truth from the average person.

There are many ways to use source criticism to reach conclusions that allow for the standing on solid ground. This is only one example.

I'm going to go a little more practical in my response to you. Think of it as the practical side of what agentdcf and itsallfolklore are saying.

There are several questions that historians ask when they approach a source. I think of them as waves that go deeper and deeper into the source. The first wave is pretty superficial:

1. What am I looking at? Is it a book or inscription or narrative art? If it is a book, what genre is it? Biography, history, epic? These all deal with the same topics but in different ways.
2. Who wrote this? Sometimes the author signed it. Sometimes the author did not. This means you have to answer with "a government official" or "a monk" or "some dude on the internet."
3. When was it written? Is it contemporary to the event/person it describes or later? How much later? This question relies a lot on context. Is there anything in the text we can use to date like the dates of a reign or an astronomic event? The books of the New Testament are dated because some reference the destruction of the Temple, is there any clue like that?

The next wave gets more at the author and his intentions.

1. Who is the audience for this? Do they have a vested interest? Does the author have a good reason to mold the story to fit its reception?
2. How was is received? Did people like it? How many copies are out there?
3. What was the author likely to know about the topic? Does he name his sources?

There are plenty of other questions you could ask about a source, but these questions can help a historian figure out how they will approach each source. I will also add that some historians answer these differently for the same source. Some may argue about the audience or the reception or even who the author was. These questions open many more avenues for study than simply asking if a source is true. Even if a source is not reliable, it can still be used to find out more about the author and his audience by asking these questions.

Sources are also true/false for given values of true, so you use them differently depending on what you're studying/doing with the info. So a first person source (like a diary) may be a very "true" accounting of one person's experiences, but horribly inaccurate as far as a broader picture goes. Or a newspaper may have exaggerated a particular story to draw certain readership and/or image of something, so the story is "false" but the reasoning behind the exaggeration says more about the general attitudes and culture of a region than would a different source. Even the conspiracy nuts, or religious fanatics, or tall-tale tellers reflect the culture in which they are participating - think about the similarities and differences in popular cultural horror that are created in Victorian Gothic literature (Frankenstein, Dracula, Poe...) and 1950's B-movies. Each one says something specific about what references the audience would share and what cultural changes were happening at the time.

So I use different information to inform me about different things, and to convey information in different ways - a diary story can illustrate and make personal the wider fact that many people came to the American West during Westward Expansion... and hated the experience, and left. Or that it was much more empty then compared to now, and the way the landscape in a particular region looks different. A newspaper story that's blatantly wrong from today's perspective is a really good way of introducing students to the idea that today's news... may be found to be just as wrong or biased or misfounded in a hundred years.