Let's take an example of using Scripture to explain Scripture, and how it can help us understand what a given text means.
Let's start with an often puzzling little passage in Revelation (only an illustrative example, not full explanation of a point of doctrine, so just two sentences). Revelation is tricky; even when using a very literal hermeneutic, and taking as much as possible at face value, there are parts that clearly must be only imagery. Sometimes, it's not obvious what an allegory/image refers to.
Revelation 12:4b-6 (ESV)
And the dragon stood before the woman who was about to give birth, so that when she bore her child he might devour it. She gave birth to a male child, one who is to rule all the nations with a rod of iron, but her child was caught up to God and to his throne, and the woman fled into the wilderness, where she has a place prepared by God, in which she is to be nourished for 1,260 days.
We want to know who the "woman" in this passage is. We could start guessing: Eve? Mary? Elisabeth? An angel? With just "a woman", a non-Christian probably couldn't make any sense of it at all. Some people with exposure to Christian culture might start naming the most popular Biblical females known to have born children.
But what does Scripture say on the matter? We need to start digging for something about a woman, and a child who rules with a rod of iron.
At first glance, Mary seems to fit the bill; she bore a child who became a King. But notice that when the child was born, he was caught up to God's throne (that didn't happen when Christ was born!). Here's our first clue that the birth imagery may be more than just Christ's physical birth from Mary. Next: this child is to rule with a rod of iron -- not newborn child's play, not a manger scene; he has an iron rod, sits on a throne, and rules over the dragon. These are clues that show we need a wider perspective. Also, after he is caught up and rules, the woman flees to the wilderness.
So what Scripture can we use to help explain this? It turns out the Revelation bit uses the same wording as a bit of a Psalm (as well as many other texts).
Psalm 2:9 (ESV)
You shall break them with a rod of iron
and dash them in pieces like a potter's vessel.
From the context of the rest of the Psalm, we can learn that this king with an iron rod is the Christ who rules not just the nations at the BC/AD switch, but all nations of the earth at all times. His authority stands as a warning to all kings.
This passage helps us understand who this king is, but what about the woman? For that, we need more information on the 'being caught up to heaven'.
Acts 2:32-36 (ESV)
This Jesus God raised up, and of that we all are witnesses. Being therefore exalted at the right hand of God, and having received from the Father the promise of the Holy Spirit, he has poured out this that you yourselves are seeing and hearing. For David did not ascend into the heavens, but he himself says, “‘The Lord said to my Lord, “Sit at my right hand, until I make your enemies your footstool.”’ Let all the house of Israel therefore know for certain that God has made him both Lord and Christ, this Jesus whom you crucified.”
In the bit I italicised, Peter himself quotes Psalms 110:1, a verse Jesus Himself used to describe Himself in Matthew 22:44!
From Peter's sermon in Acts and the Gospels' narratives, we can easily piece together that the time when Christ came into his inheritance as King was... when God raised him from the dead, and caught him up into heaven, and sat him down on the throne. This event is also described as a birth:
Colossians 1:18b (ESV)
He is the beginning, the firstborn from the dead, that in everything he might be preeminent.
Thus the "birth" referred to in our Revelation verse is not really a manger scene, but when Jesus is raised from the dead, caught up to heaven, and commences his rule.
Now we can begin to understand that the woman referred to here is the Church. While Mary was also in her own way a representative of the Church, she is just one more member of that body -- and the big picture isn't about her, it's about the entire body of believers who have been waiting for this child to be born since the promise was given to Eve in Genesis 3:15.
Without other passages of Scripture to connect it to, the scope of our sample verse in Revelation would not be clear. Where does the symbolism end, and reality begin? Is this a legend about a female human and a dragon? Are we looking for a Messiah who was supernaturally abducted as a child -- or does a man of 33, who was raised from the dead and disappeared into the clouds, fit the bill? Only using other passages of Scripture to elucidate are we able to make sense of the scene.
This is true for nearly any passage of Scripture. We saw that Peter understood the prophecy in Psalms to have been fulfilled in Christ's resurrection. Knowing how NT authors used OT passages helps us understand how we too should use them -- and when in doubt about NT passages, finding other passages that quote each other helps us accurately paint the larger picture.