What do Subscripted numbers in an equation mean?

In this case the subscripts tell you which term of the sequence you’re looking at: $F_n$ is the $n$-th term of the sequence. This particular sequence is the Fibonacci sequence, which is defined by setting $F_0=0$ and $F_1=1$, thereby establishing the zero-th and first terms, and defining the rest recursively by the relationship that you quoted in your question: $$F_n=F_{n-1}+F_{n-2}\tag{1}$$ for all $n>1$. The formula $(1)$ then says that the $n$-th Fibonacci number is the sum of the $(n-1)$-st and $(n-2)$-nd Fibonacci numbers. When $n=2$, that says that $$F_2=F_1+F_0=1+0=1\;;$$ then when $n=3$ it says that $$F_3=F_2+F_1=1+1=2\;,$$ when $n=4$ it says that $$F_4=F_3+F_2=2+1=3\;,$$ and so on.

In this way we have an infinite sequence $\langle F_n:n\in\Bbb N\rangle=\langle0,1,1,2,3,5,8,\dots\rangle$. In general $\langle x_n:n\in\Bbb N\rangle$ is an infinite sequence $\langle x_0,x_1,x_2,x_3,\dots\rangle$, the subscripts indicating the position of each term in the sequence. In the sequence the order matters. That is, although the sets $\{x_0,x_1,x_2,x_3,\dots\}$ and $\{x_1,x_0,x_3,x_2,\dots\}$ are identical, the sequences $\langle x_0,x_1,x_2,x_3,\dots\rangle$ and $\langle x_1,x_0,x_3,x_2,\dots\rangle$ are not.

You can think of these subscripts simply as labels to keep the positions straight, just as we can use $\langle x_1,x_2,x_3\rangle$ for an ordered triple representing a point in $3$-space. From a more formal point of view, however, a sequence is actually just a function. For example, the sequence $$\langle x_0,x_1,x_2,x_3,\dots\rangle$$ of real numbers is a shorthand for the function $$x:\Bbb N\to\Bbb R:n\mapsto x_n\;,$$ so that we could just as well write $x(n)$ as $x_n$.

Subscripts, like superscripts, have many uses. In this case, we have a function, probably with domain the non-negative integers or the positive integers, and $F_k$ is an abbreviation for $F(k)$.

So you can read the equation as $F(n)=F(n-1)+F(n-2)$. The value of the function at $n$ is the sum of the function values at $n-1$ and $n-2$.