The t-test is a method of answering the question, "What is the probability the data looks as it does if the true mean equals the hypothesized value?" The t-test does two things. First, it lets you avoid having to know calculus. Second, it rescales the problem so that there is one answer to the question rather than an infinite number of answers.
There are several variants of the t-test, depending on what you want to be able to do. It is actually one of the more robust tests out there in that it can be used for a wider variety of purposes than most people think about.
The t-test itself does exactly what integral calculus does, it tells you how much area is under a curve over an interval. You look up the value from the t-test and it gives you the area from 0 to 1. That area is the p-value. The value from the t-test is a rescaling of the problem from the observed values to a standard value that you can use a table to look up the solution.
The p-value is the probability that the hypothesis you tested is false. What makes the p-value important is that it is the method you can know something is false. Do you have a belief about the world that can be tested as false, if so this is one way to do it.
Prior to doing the experiment, you need to decide what p-value you will accept. For example, if you do not trust your data then you will set a very high p-value. Likewise, if it is not very important you can set a low p-value. A p-value is the probability that the data would have aligned itself as it did actually happen, if you were correct about the way the world works. Custom usually sets p-values, which is rather unfortunate. There is a trade off between false positives and false negatives.
The t-test tells you the p-value by using that value to look up the p-value in a table.
Remember, the purpose of the t-test is to tell you the probability(data|hypothesis). It does not tell you the probability(hypothesis|data). The p-value is the probability the data is as it is if you understand the world correctly.
There is a different branch of statistics, which you will not learn in this course, which studies the opposite question. Bayesian statistics studies, "what is the probability the hypothesis is true given the data is as it is?" Most people actually want the second question answered and not the first. Most disciplines tend to use the long run frequency based methods for reasons that are beyond the scope of this first course you are taking.
A low p-value say 5% would mean that there is a 5% probability that the data could look like this, if your understanding of the world is correct.