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Let $X$ be a random variable which simulates a dice rolling. We all know that the expectation (or the mean) of $X$ is : $$E[X]=\frac{1}{6}(1+2+3+4+5+6)=7/2$$

Does this mean that the expected value which will be observed after rolling the dice is $7/2$ (it does not exist for sure)? Or what it means?

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    The expectation, in my opinion, has a confusing name, since it can lead to this situation. It is the average of the expected values, meaning that if you repeat the experiment $n$ times, and $x_1, \cdots, x_n$ are the results $$\frac{x_1+\cdots+x_n}{n}\rightarrow E[X]$$ in some sense. – 2017-02-06

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It means that if you roll the die many times, and if you take the mean of your rolls, then we expect it will be around $7/2 = 3.5$.

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    You just roughly formulated the Law of large numbers. It is only one of the several reasons why expectation is important. Expectation just appears to be convenient in probability theory, but sometimes expecting that your mean will actually be near expectation is misleading, see https://en.wikipedia.org/wiki/St._Petersburg_paradox – 2017-02-06