Linea Algebra: I do not understand this definition of a Vector Space I'm studying linear algebra, I'm reviewing Vector Spaces, and i came across the definition of this Real Vector Space $( \mathcal P_n (\mathbb R), +, \cdot\mathbb R)$:

where the 'addition' and 'multiplication' operations are defined as follows:

And I just don't know how to read these definitions, i.e. what do they mean
(of both, the real vector space and the operations)

The main confusing thing in this definition is "$x$" and its "powers" $x^k$, which are not introduced anywhere, and strictly speaking do not stand for anything in particular. So you should consider the vector "$a_0+a_1x+a_2x^2+\cdots+a_nx^n$" merely as an expression, in which the numbers $a_i$ can be varied but everything else (plus signs and the symbols $x^k$) is fixed, just like in the vector $(a_0,a_1,a_2,\ldots,a_n)$ of the vector space $\Bbb R^{n+1}$ the numbers$~a_i$ van be varied, but the parentheses and commas are obligatory.
In order to have a vector space, one does need to define what addition and scalar multiplication, which is what the second image in the question is somewhat clumsily trying to do. What this amounts to is saying that the vectors in $\mathcal P_n(\Bbb R)$ are added and scalar-multiplied exactly as vectors in $\Bbb R^{n+1}$ are, but adapting for the different notation. So the second line of the image is just the counterpart of the definition
$$
  (a_0,a_1,a_2,\ldots,a_n) + (b_0,b_1,b_2,\ldots,b_n)
 = (a_0+b_0,a_1+b_1,a_2+b_2,\ldots,a_n+b_n)
$$
of addition in $\Bbb R^{n+1}$, but changing the "decoration" from parentheses and commas to symbols $x^k$ and plus signs. This does cause potential confusion, since many of the plus signs are just dressing up of expressions, while others do correspond to actual addition of vectors (being defined) or of scalars.
All this would be quire weird if there were not some uses of elements of $\mathcal P_n(\Bbb R)$ beyond the pure vector space setting. The "polynomial" notation is suggestive, and allows some abbreviations such as writing $x^2-1$ or even $(x+1)(x-1)$ for what really is $-1+0x+1x^2+0x^3+\cdots+0x^n\in\mathcal P_n(\Bbb R)$. Also one will probably want to consider the operation of replacing $x$ throughout the expression by a number$~a$, and then interpreting everything as real arithmetic operations (including the $x^k$ becoming the $k$-th power $a^k$ of$~a$), so that the whole expression is turned into a simple numeric value; this operation is called "evaluating  the polynomial at $x=a$". This operation is not part of what one can generally do in vector spaces, but it is something one can do in $\mathcal P_n(\Bbb R)$ in particular.
