The truth table for logical implication can be confusing until you realize that it is really designed to be used together with quantification. The general scheme is something like this:
Imagine that we're attempting to consider a number of different situations at once (since in mathematics we always try to save work by doing the work form many cases only once). The different "situations" may be different points in time, different places, different parameters to a function, different functions or (more vividly but less concretely) different "possible worlds".
What we then often want to do is to express a statement of the form "Every situation that has the property $P$ also has the property $Q$". For example: "Every day when I am hungry is a day when I eat". That statement only claims something about days when I'm in fact hungry -- it might still be the case that there are non-hungry days where I nevertheless eat; that does not contradict my statement about the hungry days.
Now, such a claim "in every situation where $P$ holds, $Q$ holds too" is a bit cumbersome to reason formally about, because then we need to do reasoning that applies to only some situations, and we'd need a mechanism for keeping track of which situations we're considering at a particular point in the argument. The way we handle that is by a trick by which we transform our statement to one that is about every situation. Start with
On every day when I'm hungry I eat.
Trick, part one: Instead of claiming that this is true, claim that it is not false:
It is not the case that I'm sometimes hungry without eating.
Or, explicitly mentioning days again:
There is no day such that on that day I'm hungry and on the same day I don't eat.
Part two: Note that "there is no day such that $X$" is the same as "every day is a not-$X$ day".
On every day it is not true that (I'm hungry and I'm not eating).
Now we've expressed our original claim about some days as a statement about every day, where the thing that must be true on every day is built from "I'm hungry" and "I'm not eating" using AND and NOT connectives, which I assume you're already familiar with. We can do this in general: Instead of "In all situations such that $P$ it also holds that $Q$", we write: "In all situations it holds that $\neg(P\land \neg Q)$". Then we don't need any specific logical machinery to reason about "all situations such that", but can make do with knowledge about AND and NOT.
Now it turns out that this construction is so useful and common that it is very convenient to have an abbreviation for $\neg(P\land \neg Q)$ so we can recognize the standard construction easily without needing to check that all the $\neg$s and $\land$s are in the right place. You know the abbreviation already: it is "$P\to Q$", and you can check that the truth table of "$P\to Q$" is indeed the same as that of $\neg(P\land \neg Q)$.
It only remains to justify the convention that we pronounce $P\to Q$ as "if $P$ then $Q$". (Note that this is really only a pronunciation convention: the meaning of $\to$ is whatever its truth table says it is, regardless of pronunciation). Ultimately, this is just a convention that one has to learn.
This convention does match how "if..then" is sometimes used in everyday English -- for example you can say "if the report is not done by Friday, then I will fire you", and it may happen that the report is done on Friday and you fire the employee anyway (say, because he assaulted a coworker), which doesn't make a lie of your earlier threat to fire if the report didn't get done.
But there are other ways to use "if...then" in Engliosh that do not match the logical semantics of $\to$. That is neither a problem for English nor for logic, unless you fool yourself into thinking that the two ought to be the same.