Why did they lay palm branches in front of Jesus?

After Jesus raised Lazarus from the dead in his home in Bethany, He headed for entry into Jerusalem. The people believing He was the true Messiah and would only come once as a conquering hero, laid the road with clothing and Palm branches to honor their King. They were probably quite shocked to see Him riding on a donkey instead of a warrior's horse but this fulfilled the prophecies of old. He would be tried and put to death by crucifixion mid-week (3 1/2 days). This too fulfills prophecy as He would come the 2nd time in power and might after Satan's failed attempt to conquer and control humankind in 3 1/2 years before the Millennial Rule of Christ and the Church of God.
The people of Israel laid down palm branches out of reverence for the Lord Jesus. They were welcoming Him to Jerusalem.
:John Shelby Spong (Jesus for the NonReligious) puts it this way:
The Jewish eight day celebration of the harvest, known as Sukkoth, and also called the Festival of the Tabernacles or Booths, was probably the most popular holiday among the Jews in the first century. In the observance of Sukkoth, worshippers processed through Jerusalem and in the Temple, waving a bunch of leafy branches made of willow, myrtle and palm. As they waved these branches in that procession, the worshippers recited words from Psalm 118, the psalm normally used at Sukkoth. Among these words were "Save us, we beseech you, O Lord." Save us in Hebrew is hosianna or 'hosanna'. This is typically followed by "Blessed is the one who comes in the name of the Lord. (Psalm 118:25-6)."

Although the Passover is too early for leafy branches (except palms), Mark 11:8-9 (NAB) says when Jesus entered Jerusalem, "Many people spread their cloaks on the road, and others spread leafy branches that they had cut from the fields. Those preceding him as well as those following kept crying out: 'Hosanna! Blessed is he who comes in the name of the Lord!'." The Gospels of Matthew and Luke more or less follow Mark, but John's Gospel corrects this to say 'palm branches', creating our modern tradition of Palm Sunday.