To elaborate on what has already been said, the notation $F(x) = \int_a^x f(t) dt$ means the following: sketch a graph of $f$, and find the area under the part of the graph that starts at an input value of $a$ and ends at an input value of $x$. The area you compute is $F(x)$.
For example, you can figure out that $\int_1^x 2t dt = x^2 - 1$ by just sketching the function and using the formula for the area of trapezoids. So in ths example, $f(x) = 2x$, and $F(x)$ in part 1 of the theorem has to be the function $F(x) = x^2-1$.
The second part of the theorem says that if you want to find $\int_a^b f(x) dx$, that is, the area under the graph of $f$ starting at an input value of $a$ and ending at an input value of $b$, you can do the following: choose any antiderivative $G(x)$, for $f(x)$, and then take $G(b) - G(a)$. The result is the area you seek to find. For example, suppose you want to find $\int_2^3 x^2 dx$. Again, $f(x) = 2x$, but you can take $G(x)$ to be $x^2 - 1$ or $x^2 + 3$ or $x^2$, since all of these functions have $2x$ as their derivative. No matter which one of these you choose, $G(3) - G(2)$ will give you the area under the graph of $f(x) = 2x$ from $x = 2$ to $x = 3$.