What is the intuitive meaning of Expected Value? What exactly does this number mean? I am aware on how to calculate the them given the variables. I have watched several videos and articles about it but no one seems to explain it. Maybe I am just missing something and it is just a meaningless number?
I watched this example about a survey of students that rate their class overall satisfactions. 1 - very dissatisfied and 5 very satisfied. x = 1,2,3,4,5 and the count in the order of x is count - 5,10,11,44,38 = 108 After calculating the E(X) = 3.7 I noticed it is weighing more towards the very satisfied side. What exactly is this number telling us aside from being just an average?
 A: The expected value a.k.a. mean is one of several measures of "central tendency," which is, loosely speaking, a number that is representative in some sense of the values that samples from the distribution take. Although it corresponds to the average value (add them up and divide by the total number) that we learn to take as children, it is a bit obscure what it actually is. The other most popular measures, the mode and the median, are easier to articulate succinctly: the mode is the most probable value and the median is the value for which there is an equal probability of being above and below.
The mean is a bit more elusive. Sometimes it is a fairly typical value (or close to a fairly typical value) but other times it's not. There are heavy-tailed distributions in which the mean is much larger than the median or mode, so that the mean is not a very typical value you'd expect to see if you drew as sample from the population at random. For some distributions, the mean is not even defined.
Here's a classic example. Say we play a game where we roll a pair of ten-sided dice. If both the dice are one, you win and I give you 100 dollars. If it comes up anything else, I win and you give me one dollar. The probability that you win is very low, only $1/100,$ so you are overwhelmingly likely to lose and pay me a dollar. Your median and most-likely outcome is to lose a dollar here. However your average outcome is $$ \frac{1}{100}\times \$100 - \frac{99}{100}\times \$1 = \$0.01.$$
Even though you lose overwhelmingly often, your average outcome is positive. The reason, of course, is that when you do happen to win, you win big. The mean incorporates this effect while the mode and median do not. 
This example shows you one way of understanding what the mean is. If you play this game over and over many times and compute your average earnings (total earnings / number of games played), then as you play more and more, your average earnings will approach $\$0.01$ per game. (See law of large numbers.) This tells you that while playing this game might be bad for you and good for me in the short run, in the very long run you will actually win a bunch of money from me. It will be slow and painful, and you will lose many more games than you win, but the big payments from the few you win will be more than enough to compensate. In fact, if we played this a million times, you would typically win something like $\$10000 \pm \$100$ and there would be a negligible chance of losing money in total.$^*$
Another classic example is wealth. If you approach a random person and ask them their wealth they will probably be close to the mode, the typical amount of wealth for a person to have. However it's well known that the very, very rich have hundreds or even thousands times that. A typical rich celebrity might have something like 40 million dollars. Warren Buffet, the richest person in the world, has $80$ billion dollars, two thousand times that. By contrast to both, a typical person you'd meet walking around might have $\$40,000.$ A thousand times less than the celebrity and two-million times less than Buffet. 
If you looked that the average net worth of people in the US, you'd see it's a lot more, perhaps something like $\$400,000$, which is ten times the median. This means that the very rich being so much richer than a typical person are having a big effect that is dragging the mean upward. (I should emphasize these numbers are just made up by me, although I glanced at some statistics to make sure they weren't wildly off.)
This second example shows another way to think about the mean. If you took all of the money from everybody - the poor, the typical, the very rich and the mega-rich, and then evenly distributed it to everybody, we'd all have the mean wealth. 
A third example comes to mind. Since the mean is supposed to be representative of the values, it makes sense that it would be required to be "close" to all the values in some sense. In fact, the mean of a sample $x_1,x_2,\ldots,x_n$ is the number $\bar x$ that minimizes $ (x_1-\bar x)^2+(x_2-\bar x)^2+\ldots+(x_n-\bar x)^2,$ the sum of squared distances to the sample points. On the same note, as bof mentioned in the comments, the mean is analogous to the center of mass if you think of each sample $x_i$ as an equal-sized mass at location $x_i$.
$\;^*$For an example of a game where a typical amount to win is modest but the mean is infinitely large, look up the St Petersburg Paradox.
A: From your example, it's the average of the samples, because it's based on a survey ("after-the-fact" distribution).
One other way to think about it is, as the term suggests, what do you "expect" when you sample this random variable.
For example, if you throw a fair dice and get the following reward: if an even number shows up, you get 10 dollars. If 3 or 5 shows up, you get $\$$20. If 1 shows up you lose $\$$-40. So what's your expected earning when you throw this dice? Meaning, what can you expect out of doing this dice? If you do the calculation, you get $\$$5 as the expected value. So if this were a carnival game, would you pay $\$$1 to play this game? Or would you pay $\$$4? What about $\$$10?
