Assuming that a $\bot$ indicates we have two contradictory statements $A$ and $\neg A$, according to wikipedia this can be the reasoning behind $\bot\vdash B$:
Now is my assumption correct and hence this the logic behind $\bot\vdash B$? If not how does this rule make sense?
As discussed in various posts, there are slightly different presentations of the calculus of Natural Deduction: here is a "good" one.
For a more detailded treatment, you can see the beautiful book:
The differences are mainly limited to the "management" of the negation sign (see e.g. this post ).
The above set of rules can be "streamlined" further, avoiding $\lnot$ and introducing it as an abbreviation for $P \to \bot$.
With this decision, we can remove the two rules: $\lnot$E and $\lnot$I because they are simple consequences of $\to$E and $\to$I and the abrreviation.
$\bot$E (ex falso) is needed for Intuitionistic Logic. If we remove $\bot$ and the $\bot$E rule, we get the so-called Minimal Logic.
As discussed elsewhere, ex falso and modus tollendo ponens are inter-derivable; but the use of the last one as primitive is contrary to the "spirit" of ND, where every connective (except for $\bot$: the lack of an $\bot$I rule is clearly motivated by the need to avoid the inconsistecy of the calculus) is "defined" by a couple of rules: the intro- and the elim- rules.
Mtp instead, uses both $\lnot$ and $\lor$.
End of the "resumé".
Regarding $\to$-intro, I'm using the case: $A \vdash B \to A$.
This is perfectly sound: if $A$ is (assumed to be) true, then $B \to A$ is also true, under assumption $A$; see von Plato, page 33.
This "intuition" can be formalized through the fact that we can clearly derive $A$ from assumption $A$. But we can always add "unnecessary" assumptions; thus, we have also $A,B \vdash A$ and so, by $\to$-intro: $A \vdash B \to A$.
But the last explanation amounts exactly to the "insertion" in the derivation of a new assumption and thus the technique, of "re-iterating" $A$ under assumption $A$ (as you have done in your post) is a legitimate application of it.
In the above comments, I've played with the abbreviation:
1) assumed $\bot$
2) then: $\bot \vdash \bot$ --- by reiteration
3) $\vdash \bot \to \bot$ --- by $\to$I, discharging 1)
4) $\vdash \lnot \bot$ --- by abbreviation
and more: 5) $\vdash \top$ --- the TRUE: a new abbreviation.
We have proved $\vdash \top$, and this is consistent with the "completeness" of the calculus: we want to prove all formulas that are always true, and the TRUE constant is one of them.
In Hilbert-style proof systems we can use $\to$ and $\lnot$, like Frege, or $\lor$ and $\lnot$, like Russell-Whitehead. Or, see Tarski-Bernays and Church, we can have $\bot$ as primitive.
Again, some variations: Tarski uses the axiom: $\bot \to A$, while Church uses Double Negation: $((A \to \bot) \to \bot) \to A$.
And more... we can "fill the gap" with ND using a couple of axioms for every conncetives.
