Intuitive understanding of the t-value A manufacturer claims that the fuses blow less than 10 minutes on the average. You take a sample of 20 fuses and record the sample mean as 10.4. The sample standard deviation as 1.6 minutes. The sample is from a normal population. Does the data support or refute the manufacturer's claim. 
My doubt: Calculating the t-value is easy, you get 1.118. My doubt is that the book states the answer as "the probability that t exceeds 1.328 is 0.10 and because the observed value is less than 1.328 and 0.10 is a large probability, we can claim that the manufacturer is correct."
I don't understand how to interpret this statement or understand what the calculated t-value (1.118) actually means. 
 A: Let's start my writing down our null and alternative hypothesis.
The null in this cast is that the time for the fuse to blow (which I will denote $w$ since we will use $t$ later) is less than 10 minutes
$$ H_0: w\leq 10 $$
The alternative is 
$$ H_A: w>10$$
You've correctly computed the t-statistic.  That is
$$ \hat{t} = \dfrac{10.4-10}{1.6} \sqrt{20} \approx 1.118$$
Since we are doing a one sided test, we should evaluate the probability that the data we observe are at least this large or larger.  That would require us to (if I were teaching an undergrad stats class) go into our t-table, find the column for the appropriate degrees of freedom (19 in this case) and the row for 1.118.  Chances are the table does not have 1.118 exactly, so what the author has likely done is chosen the nearest value which is larger than 1.118.  In this case, that is 1.328.  The author astutely notes that since there is a 10% chance of seeing a t-statistic as large as 1.328 or larger, then the probability we see a t-statistic as large as 1.118 or larger must be larger than 10%.  Since we often call probabilities smaller than 5% "statistically significant", we can be conclude that the data supports the manufacturer.
OK, but what is the t-statistic?  That is a much harder question to answer.  Let's start with our hypotheses.
Our null hypothesis is that the time for the fuse to blow is less than 10 minutes.  That is our present belief about the world.  We can never confirm that belief, we can only find evidence against it (that will become important later).
So we collected some data on time for fuses to blow and computed a mean and a standard deviation.  We know that the data are random, but we can actually consider the mean and the standard deviation to be random quantities too! Why? Well, the data that comprise the mean and standard deviation are random, and if I were to perform the experiment again, then my mean and standard deviation would be different than the mean I got from this experiment.
Through some math which I won't get into, we know what the distribution of the mean divided by the standard error (that is the standard deviation divided by the root of the sample size) would be if the time for the fuse to blow really was less than 10.  What we do when we compute the t statistic and the associated probability from the table (or from R or whatever) is essentially tell ourselves...

Right now, I believe that the time for the fuse to blow really is less than 10.  If that were true, then the probability that I would have observed fuses blowing as long or longer than I presently observed is $p$.

Here, $p$ is the p value.  If that probability is really small (5% in most cases) then one of two things has occurred.  Either...
1) Our initial belief about the time for fuses to blow is wrong, or
2) We have observed something incredibly rare.
We often believe 1) over 2), and that is why we reject the null hypothesis.  Essentially, we believe that seeing fuses blow at least as long as we presently observed is not consistent with the belief that fuses take 10 minutes to blow. 
