The .05 significant level has nothing magical to it and is taken by tradition. I suppose you know that the Null Hypothesis states that there is no difference between samples. We start out by assuming it is true. Let's take an example without a Significance Level at first to illustrate so that you can understand it intuitively.
Let's say I have a friend who claims he has a super power and he can predict whether a coin shows heads or tail in my hand even before peaking at which side it shows. Let's say he specifically claims he can We decide to put his theory to the test.
A coin toss is a Binomial Event: after each coin toss he can either succeed or fail. We agree to toss the coin $n = 100$ times. The probability of him getting it right (or wrong) at each toss is 0.5, and since the Expectation for the Binomial Distribution is computed as $E[X] = n.p$ then the expectation in this case is 100 x 0.5 = 50. This means that we expect a person that does NOT have any super power to get 50 "predictions" right just out of pure luck. This means, for us to be ready to admit he may indeed have some super powers, he needs to beat this number (i.e. do much better than what the average person with no powers can do out of simple luck). And you can see that the more he exceeds that Expectancy, the more we can concede that he did something very unlikely that cannot be explained solely by luck, and that he may indeed have some powers, thus we can be satisfied enough to reject the Null Hypothesis and assert that yes, there IS a significant difference between his performance and the average performance.
Let's say this friend of ours managed to correctly predict 54 coins. He's so happy about his performance and he tells us that that's more enough evidence for his powers. But you can see is not that far from 50, which is what we expect a normal person with no powers to achieve. If you compute the probability of getting 54 correct calls (or more) with the given parameters, you'd find that by pure chance, a person has almost 0.2 probability of achieving this performance (of 54 calls). That's a high number. Meaning a person with no powers can achieve this same performance by pure chance. And so him managing to beat the Expectancy can be entirely explained by chance, and so we don't have enough evidence against the Null Hypothesis: we reject the Alternative Hypothesis.
Let's take another case wwhere he managed to predict 68 coin tosses. Computing this using the Binomial Distribution we get around 0.0001 probability: this is a VERY extreme event. An extremely rare occurence that is very unlikely to happen, and yet it did. The way we view this is: assuming the Null Hypothesis is true (assuming the friend has no powers as a start) the probability of an average person with no special powers to achieve this is 0.0001. And seeing that an average person is extremely unlikely to achieve such performance and thus it cannot be entirely attributed to little variations around the expectation due to pure luck, then we are ready to concede that there is enough evidence against the Null Hypothesis, and so we reject it.
This is the intuition behind the p-value. You asked " If the p-value
of the test comes out to be a small number, this fact is taken as
justification for "rejecting the null hypothesis." Why is this a
reasonable conclusion? ".
Well in our coin toss example, the p-value = 0.0001. This is a very extreme probability, and so you can see why this probability being small would give us evidence against the Null Hypothesis. This p-value being smaller and smaller means our friend performs better and better: he achieves more and more extreme and very unlikely events that cannot be attributed solely to random variations due to luck alone. The smaller the p-value the more our friend's performance moves to the extreme right of the distribution. And since by definition the Null Hypothesis is the assumption that there wouldn't be any difference between his performance and the average expected performance, then him moving more and more away from the mean gives stronger and stronger evidence against the Null Hypothesis, and that's why small p-values make us reject the Null Hypothesis.
Finally, about the Significance Level, as I said that's just by tradition. There's much debate about what's an appropriate level. In our example, 0.0001 is a lot smaller than 0.05 and thus if we were using that level our Null Hypothesis would've been rejected.