This is a great question that digs at the heart of how we use probabilities. There are several great answers, but I'd like to offer one which is more intuitive in nature.
The key to understanding the issue you are dealing with is understanding your assumptions. Let us say that you have a coin which, when flipped, lands heads 50% of the time and tails 50% of the time. This is a fair coin. You flip this coin 100 times and it lands on heads every time. It still has a 50/50 chance of landing tails on the 101th toss. That's simply how probability works.
Now let's explore your case. In your case you have a coin which has never been flipped. You then state
Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5.
This is an assumption. You are tying the real life coin to a statistical distribution, assuming you understand how it will behave. You did not state why you feel this assumption is justified, but presumably this is because you have some experience with the physics of how objects called "coins" tend to behave when flipped. You may even have a mathematical model based on a theory of how the material in the coin is distributed.
You now flip the coin 100 times. What was the expectation for the number of times it lands on heads? Well, we don't actually know. However, we have made an assumption that this flipping process will yield 50/50 odds, so you expect 50 heads. You can also then use the Central Limit Theorem to determine the variance you expect, which will be quite narrow for 100 tosses.
Then your coin actually lands on heads 100 times. What now? Nothing in the probabilities stated a fair coin couldn't land on heads 100 times. Indeed, probability suggests that one should expect that occurrence. It would be rare, but probability would state that it should happen now and then.
You made an assumption earlier: that the coin was fair. This was justified in some method, typically by an assumption that you know how coins are flipped. Rationally, we want to think of these assumptions as sacrosanct, but intuitively you know something's going on (after all, you asked a question on Mathematics.SE about it!). So let's take a rational look at it. We made an assumption, perhaps that assumption was wrong. Perhaps the coin is magnetic and we are in a magnetic field. Perhaps the coin is weighted strangely. Perhaps its just dumb luck and our assumption was actually valid.
At this juncture, statistics and probability provide two major approaches for dealing with this rational conundrum: frequentist and Bayesian. Both of these schools have good answer with lots of points on this question. The frequentist approach is to explore whether we should reject the assumption that the coin was fair to begin with. One gathers data points (such as the 100 tosses) and then makes decisions. One may decide that the probability of 100 heads coming from a fair coin from random chance is low enough that you wish to reject the hypothesis that the coin was fair in the first place. That wording is careful. It doesn't state that the coin is fair, merely that it rejects an assumption that it was fair because the data challenges the assumption.
The Baysians take a different approach, and instead try to continuously improve their assumptions. They start with the assumption that the coin is fair. Perhaps more precisely they assume that the actual distribution is very close to that of a fair coin. They then begin flipping the coin in the test. Each time they update their assumptions about the probability distribution of the coin. As they do, they develop a hypothesis that the coin is more and more unfair as they see more and more heads.
Both approaches have their merits. There's people in each camp. However, the thing they both agree upon is that the assumption that the coin is fair is an assumption worth challenging. The frequentists try to reject the hypothesis, the Bayesian try to adapt it into a more correct hypothesis, but neither camp tries to hold themselves to the assumption that the coin is fair.
Because the world isn't fair.