Conditional Statements: "only if"

Think about it: "$p$ only if $q$" means that $q$ is a necessary condition for $p$. It means that $p$ can occur only when $q$ has occurred. This means that whenever we have $p$, it must also be that we have $q$, as $p$ can happen only if we have $q$: that is to say, that $p$ cannot happen if we do not have $q$.

The critical line is whenever we have $p$, we must also have $q$: this allows us to say that $p \Rightarrow q$, or $p$ implies $q$.

To use this on your example: we have the statement "Samir will attend the party only if Kanti attends the party." So if Samir attends the party, then Kanti must be at the party, because Samir will attend the party only if Kanti attends the party.

EDIT: It is a common mistake to read only if as a stronger form of if. It is important to emphasize that $q$ if $p$ means that $p$ is a sufficient condition for $q$, and that $q$ only if $p$ means that $p$ is a necessary condition for $q$.

Furthermore, we can supply more intuition on this fact: Consider $q$ only if $p$. It means that $q$ can occur only when $p$ has occurred: so if we don't have $p$, we can't have $q$, because $p$ is necessary for $q$. We note that if we don't have $p$, then we can't have $q$ is a logical statement in itself: $\lnot p \Rightarrow \lnot q$. We know that all logical statements of this form are equivalent to their contrapositives. Take the contrapositive of $\lnot p \Rightarrow \lnot q$: it is $\lnot \lnot q \Rightarrow \lnot \lnot p$, which is equivalent to $q \Rightarrow p$.

I don't think there's really anything to understand here. One simply has to learn as a fact that in mathematics jargon the words "only if" invariably encode that particular meaning. It is not really forced by the everyday meanings of "only" and "if" in isolation; it's just how it is.

By this I mean that the mathematical meaning is certainly a possible meaning of the English phrase "only if", the mathematical meaning is not the only possible way "only if" can be used in everyday English, and it just needs to be memorized as a fact that the meaning in mathematics is less flexible than in ordinary conversation.

To see that the mathematical meaning is at least possible for ordinary language, consider the sentence

John smokes only on Saturdays.

From this we can conclude that if we see John pulsing on a cigarette, then today must be a Saturday. We cannot, out of ordinary common sense, conclude that if we look at the calendar and it says today is Saturday, then John must currently be lighting up -- because the claim doesn't say that John smokes continously for the entire Saturday, or even every Saturday.

Now, if we can agree that there's no essential difference between "if" and "when" in this context, this might as well he phrased as

John is smoking now only if today is a Saturday.

which (according to the above analysis) ought to mean, mathematically, $$ \mathit{smokes}(\mathit{John}) \implies \mathit{today}=\mathit{Saturday} $$

I see it this way:

"If Kanti will not be at the party, then neither will Samir", which translates to $\neg q \to \neg p$ which is logically equivalent to $p \to q$.