I will attempt an answer that I hope will stick as much as possible to the text itself, avoiding the flights of fancy that are possible.
Leon Morris on the meaning of the word ἱλαστήριον in verse 25:
There are problems with the next expression, which KJV rendered “a propitiation” and which nearly every modern translation waters down.Part of the trouble is that “propitiation” is neither a well-known nor a well-used word today, and translators like to employ something simpler. But the major reason is that propitiation means the removal of wrath and, as we saw earlier (see the notes on 1:18), some commentators find the concept of the divine wrath distasteful and unworthy; so they write it out of Scripture.If there is no wrath there is no propitiation. There are two major reasons for rejecting this approach. One is the meaning of the word Paul uses. The detailed examinations mentioned in n. 126† show that the word means “the removal of wrath”. The other is the context. Paul has mounted heavy artillery in the section 1:18–3:20 to show that all are sinners and subject to the wrath of God. But unless the present term means the removal of wrath he has left them there, still under God’s wrath. Nothing else he says means the averting of wrath. On both accounts we should see Paul as saying that from one point of view Christ’s death means the removal of the divine wrath from believing sinners.
Leon Morris, The Epistle to the Romans, The Pillar New Testament Commentary (Grand Rapids, MI; Leicester, England: W.B. Eerdmans; Inter-Varsity Press, 1988), 180–181.
In Isa. 53:4-7
4 Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him stricken, smitten of God, and afflicted. 5 But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed.** 6 All we like sheep have gone astray; we have turned every one to his own way; and the LORD hath laid on him the iniquity of us all.
An article defending Penal substitution:
What we know from passages such romans 3:25 is that Jesus was our propitiation. That through His sacrifice the wrath of the Father was satisfied. That implies that during the three hours He was on the cross that the wrath of the Father was being poured out upon Him as our great substitute. It is during this time that Jesus was also forsaken by the Father (Mark 15:37, Matthew 27:46). In this view, the darkness of the three hours was the physical manifestation of God's wrath that everyone could see. It may even have contributed something, along with the other events, to the mind of centurion (Mark 15:39); where he declared "truly this man was the Son of God." To a pagan like the centurion the physical manifestations in the heavens would have surely led him to believe that the gods were angry. It was not the gods who was at that moment were angry, it was the one true God who was pouring out His anger at the Son that we might never have to pay for our sins.
The concept of penal substitution is is based on God's justice. God's justice requires that there is a required punishment that must be paid. Therefore His death is more than just His death. This not appeasement which is a pagan concept. Appeasement as a concept of the actions of a man towards a god. Penal Substitution is the plan among the members of the Godhead, It is God who places the sins of the world on God the Son (2 Cor. 5:21). This was planned before the foundation of the earth. This points out that no man was responsible for the death of Christ, neither the Jews nor Pilate. It was the Father who brought about the death of the Son and it was always His purpose for the incarnation so that God was able to forgive sinners since the penalty for their sins had been paid by the sacrifice of His Son. Any systematic theology book would give numerous references within the Scriptures to explain this doctrine. Those texts would also detail the fact that a denial of this doctrine is an ancient heresy that has been around for centuries, in addition to the many other attempts to come up with a different intent of the atonement.
The wrath of God is a common theme in the book of Romans. Numerous times Paul refers to wrath in the book of Romans and nearly all of them refer to wrath of God either directly or indirectly.
Verses in Romans (bold directly referring to the wrath of God)
Ro 1:18 For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness;
Ro 2:5 But after thy hardness and impenitent heart treasurest up unto thyself wrath against the day of wrath and revelation of the righteous judgment of God;
Ro 2:8 But unto them that are contentious, and do not obey the truth, but obey unrighteousness, indignation and wrath,
Ro 4:15 Because the law worketh wrath: for where no law is, there is no transgression.
Ro 5:9 Much more then, being now justified by his blood, we shall be saved from wrath through him.
Ro 9:22 What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction:
Ro 12:19 Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord.
Ro 13:4 For he is the minister of God to thee for good. But if thou do that which is evil, be afraid; for he beareth not the sword in vain: for he is the minister of God, a revenger to execute wrath upon him that doeth evil.
Ro 13:5 Wherefore ye must needs be subject, not only for wrath, but also for conscience sake.
As these verses show, the wrath of God already abides upon those who reject the person and work of Jesus Christ. Now if the wrath of God will be poured out upon the lost, and Jesus is our substitute then, by inference, He received the just punishment we all deserve, namely His wrath. That is the very heart of substitutionary atonement which is the very foundation of Biblical Christianity. To deny that He received our just punishment is to leave a person in their sins since they reject the person who paid the price for their sins.
Romans 3:25-26 is showing how God could be both just (righteousness) and at the same time the justifier of the ungodly. To do this we have to have faith in His substitionary sacrifice--the phrase through faith in His blood. If it was just faith in Him them there was no need to mention propitiation or His blood. His blood is in direct reference to His sacrifice. In doing so he became our propitiation (the Greek word ἱλαστήριον in verse 25).
Cross references include John's emphasis on the fact that Jesus is the light of the world. An interesting idea is also that of Revelation 21 where in the city of the new heavens and the new earth there will be no sun and moon as the glory of God will provide the light for the eternal city. So it should not surprise us that the Father removes the light during that time when the wrath of God was being poured out on the Son.