Key to answering this question is that no mainstream Christian denomination thinks they are actually worshipping any person other than the Triune Godhead - God the Father, God the Son, and God the Holy Spirit.
From the outside, however, accusations are often made in this regard.
To the uninitiated, veneration of the Saints can look like worship.
To the uninitiated, the use of icons can look like idolatry.
To the uninitiated, requesting the assistance of angels, Mary, or whatever blessed Saint can look like praying to someone other than God.
To the uninitiated, repeating the Lord's Prayer or the 23rd Psalm can look like the invocation of a talisman against evil.
And, in practice, the distinction may not actually be there for the person doing it. Some Catholics actually believe they are praying to Mary. Some Orthodox think they are worshipping their icons. And some evangelicals may believe that the Lord's Prayer is somehow "holier" than every other.
The point, however, is not to judge a theology by its worst adherents, but by its best. Any doctrine can be perverted, any good thing used for evil.
In 2 Kings 18:4, we read of what happened to the bronze serpent that God had instructed Moses to build after God had sent a plague of vipers to attack the Children of Israel in the desert. Speaking hundreds of years after the fact, the Bible records that Hezekiah:
.. removed the high places, smashed the sacred stones and cut down the Asherah poles. He broke into pieces the bronze snake Moses had made, for up to that time the Israelites had been burning incense to it.
That people had turned an artifact of God into a device of worship does not speak to the evil of a thing - only to the evil of those who practice idolatry at all. Idolatry - the worship of something or someone other than the true God - is a sin. And when "veneration" slips into worship, it is wrong.
But simple respect is not always worship. The question is, at their best, what are these things supposed to do.
In all cases, these objects, people, or beings are supposed to be intermediators to point the worshipper to Christ - a laudable goal. People need concrete things. But people are also prone, as Romans 1 says to worship the creation rather than the creator.
25 They exchanged the truth about God for a lie, and worshiped and served created things rather than the Creator—who is forever praised. 26 Because of this, God gave them over to shameful lusts. ... 28 Furthermore, just as they did not think it worthwhile to retain the knowledge of God, so God gave them over to a depraved mind, so that they do what ought not to be done
Thus, in short, the ultimate answer to "Why do people worship these things" is because man is evil. Man twists and perverts. He loves darkness rather than light.
Sure is a good thing we know a God who can fix that, if we're willing to learn differently.