On the Intercession of Saints

I want to start off by saying that I am absolutely sympathetic to Protestants who think we worship saints, because a Catholic (or Orthodox) church is extremely shocking to somebody who grew up with a typical Protestant church.
I can just imagine them thinking "Why are there pictures of regular humans in a place that is supposed to be for the worship of God alone?". Without having the proper context, I absolutely see how one could think it is idolatry.

The main root of the misunderstanding, as I see it, is twofold: the first is the distinction between honour/veneration and worship, and the second is whether or not the saints can hear our prayers.

First, then is the distinction between worship and veneration/honour. Honour is a type of respect due to all mankind, as well as to God.
The Greek word for this concept (timé), is multi-faceted, such as when one is commanded to honour their father and mother, or where Peter says "Honour [timesate] all men. Love the brotherhood. Fear God. Honour [timate] the king."
Paul says in 1 Timothy to "Honor [tima] the widows", yet we are also called to "honor [timosi] the Son, even as they honor [timosi] the Father".
So we have a clear precedent in Scripture for there being a respect or honour paid not only to the Holy Trinity, but to our parents, widows, the king, and "all men".
Thus, having established this distinction, the simplest form of the argument for honouring/veneration of the saints would be an "a fortiori" argument:
We are called by the apostles to honour all men, and we are also called to honour God. Now, it is clear that we are to honour God more than the average man.
So if there are gradations in the amount of honour which is to be paid, it logically follows that we should give more honour to the blessed Virgin Mary, than to a child rapist. For Mary bore God the Son in her very womb, and all generations shall call her blessed, how much more should we honour her than a child rapist?
Now, the idea is clear - we honour God at the top of the hierarchy in a unique way, and infinitely below him are the saints and angels in heaven who have earned their crown and robe, and below them are our brothers and sisters here on earth.

We do NOT, under any circumstances, worship saints. Worship, also called latria, is due to God alone. You can see this in such passages as Thou shalt worship the Lord thy God, and him only shalt thou serve [latreuseis]." (Luke 4:8)
Latria is a kind of exclusive worship - it is "service", in the sense of being exclusively yoked to a master. If you give latria to anybody but God, you are committing idolatry. Catholics will never, ever, give latria to any saint, even the blessed Virgin Mary. We respect and venerate the saints very highly, but NEVER worship them.

So, hopefully you see the distinction between honour and worship.
Assuming that makes sense, the segway to the next point is something like - "okay, I see that one can honour the saints in heaven in a similar way to how we honour those who are alive, and I see how it is reasonable that some particularly holy saints could be more honoured than your average person. That's all fine.
But why do you pray to the saints? Isn't prayer supposed to be directed to God alone, because " there is one God, and one mediator between God and men, the man Christ Jesus"?".

First and foremost, the word pray just means "ask". If I ask you to pray for me, I am asking you to ask [God] for something on my behalf, even though I can also pray directly to God. This is a very common Christian practice, and I will assume you already do it.
The basis for the belief of prayer to the saints is threefold: first, we are called to pray for one another to God, and asking somebody to pray for you is not wrong - second, those in heaven are aware of what is going on on earth, in a supernatural way - and third, the saints in heaven are closer to God than we are.
With regards to the first point, I'm sure you already know the verse: "pray for each other so that you may be healed. The prayer of a righteous man availeth much".
I'm sure that one would agree that IF the saints in heaven could hear our prayers, because they have already proven their righteousness, their prayers to God would be very effective.
Another a fortiori argument - if the prayer of a righteous man on earth availeth much, how much moreso would the prayer of St. John the Baptist be, seeing as Jesus Himself said of him "among those born of women there is no one greater than John", but as if that was not enough, Jesus even says "yet the one who is least in the kingdom of God is greater than he".
So IF the people in the kingdom of God COULD hear our requests for them to pray for us, and if hearing our request they would pray to God for us, it would surely be a profitable practice, because their righteousness is proven, and even the Lord calls the lowest person in heaven greater than even John the Baptist.

So, the question becomes, CAN the saints in heaven see what is going on on Earth? The scripture is as follows:
"when the Lamb opened the fifth seal, I saw under the altar the souls of those who had been slain for the word of God and for the testimony they had upheld. And they cried out in a loud voice, “How long, O Lord, holy and true, until You avenge our blood and judge those who dwell upon the earth?” (Revelation 6:9-10)
So the souls of the saints are under the altar in heaven, and they are clearly aware that God has not yet avenged their blood and judged the Earth. For this to be the case, they must know what has or has not happened on Earth.
Thus, at the very least, the saints in heaven know what is going on on the Earth. We also know that our guardian angels know what is going on on Earth in a direct, play-by-play way, with regards to each soul:
"See that you do not look down on any of these little ones. For I tell you that their angels in heaven always see the face of My Father in heaven."

So, we know that angels and saints in heaven can actually see what is going on on Earth, and that we have at least one guardian angel who both sees everything going on in our lives, and simultaneously always see the face of the Father in heaven.
Now, the next question might be, could the saints possibly hear the prayers of so many people at once? If there are over 1 billion Catholics + Orthodox, and so many of them pray to the virgin Mary, how could she possibly pray to God for all of those who are asking her to pray for them?"
The scripture is as follows:
"Then I heard every creature in heaven and on earth and under the earth and in the sea, and all that is in them, singing, “To the one seated on the throne and to the Lamb be blessing and honor and glory and might forever and ever!” And the four living creatures said, “Amen!” And the elders fell down and worshiped." (Revelation 5:13-14)
So, John the Apostle, a mere mortal who was still alive and in his physical body, was granted the ability to simultaneously hear every single living thing in the entire universe - and not only that, he was able to coherently understand them all, human and inhuman, in his own language.
The four living creatures also heard every creature in the universe simultaneously, and it is implied that the elders did too.
Thus, it is clear from scripture that there is absolutely no logistical problem with the saints and angels in heaven hearing the prayers of the faithful. God is omnipotent, and He has clearly granted this grace to at least the angels, the elders, and John the Apostle.
Therefore, there is no reason why the saints could not hear our prayers in heaven. They are aware of what is going on on the Earth, and many (if not all) of them can simultaneously hear and understand every creature in the entire universe. It would even be a small thing for God to allow His saints to hear only those prayers directed to them across the world in their own language.

Finally, now that we know the saints in heaven are aware of what happens on Earth, and that many have supernatural graces that allow them to have the ability to hear prayers, the only question is whether it is worth it to ask the saints/angels to pray for us, rather than doing it ourselves.
This should already be obvious based on the previous verses, but here is a final verseL
"And when he had taken the book, the four beasts and four and twenty elders fell down before the Lamb, having every one of them harps, and golden vials full of incense, which are the prayers of the saints." (Revelation 5:8)
So the prayers of the saints are offered like incense DIRECTLY to Jesus Christ, in His heavenly liturgy. Now, I don't know about you, but I am not a saint. But the saints in heaven are saints, and if they can hear my requests for prayers, and will pray to God for me, that means a prayer from a lowly sinner like me can be offered directly to the throne of Jesus Christ in heaven, in a way that is special and distinct from the prayers of non-saints.

Altogether, this was the convincing case which proved to me that asking the saints to pray for us is, at best, extremely profitable and worth doing (which is probably why the early Christians did it) - and at worst, contradicts nothing within scripture, and is effectively a net neutral, because we also pray to God directly.