How could Jesus get followers so easily?

Put simply, they were called by the Holy Spirit.

It's true that Jesus gave Peter some evidence, but Peter's decision of leaving all to follow Him was the inner workings of the Holy Spirit.

When John the Baptist first saw Jesus, it was not outward evidence, but the Holy Spirit that inspired him to recognize that Jesus was the lamb of God here to take away the sins of the world.

When Nathaniel met Jesus, there were no miracles except Jesus telling him that He saw him even under the fig tree. Nathaniel exclaimed "Rabbi, thou art the Son of God; thou art the King of Israel" (John 1:49)

In the old testament times, when Elijah called Elisha, it was not any outward display of miracles that made him take his hand off his father's plow and follow Elijah.

Similarly today, when we were first called by Jesus to pick up the cross and follow him, it was not sight that made us believe, but the deep conviction of the Holy Spirit.

However, not all who are called will allow conviction to take root, because the Kingdom of Heaven is like the pearl of great price, when it is found one must sell all to obtain it (Matt 13:46).

Before meeting Jesus, his disciples led boring, humdrum lives. With his "miracle," Jesus showed them that he was a competent fisherman, better than they, the "professionals." Their impression was something like, "If the Jesus fellow can beat us at our own game, what else can he do?" Without fully understanding, they felt that Jesus would take their lives to the "next level,"--and they were right.

God sovereignly appointed the Apostles.

As the Son, he chose them, as in John 15:16.

You did not choose me, but I chose you and appointed you so that you might go and bear fruit—fruit that will last—and so that whatever you ask in my name the Father will give you.

As the Father, he gave them to the Son, as in John 17:6.

“I have revealed you to those whom you gave me out of the world. They were yours; you gave them to me and they have obeyed your word.

He clarified his call as excluding Judas in John 13:

“I am not referring to all of you; I know those I have chosen. But this is to fulfill this passage of Scripture: ‘He who shared my bread has turned against me.’"

This does not rule out other miracles, like the miraculous catch of fish, the word to Nathaniel, or village stories told about Jesus' birth, temple dedication (and associated blessing), and expert grasp of scripture as exhibited his visit to the temple at age twelve. Those wold be the tangible means that backed up the sovereign call.

I once heard a preacher claim that this was one of the miracles of Jesus.

Clearly there's no biblical basis for this, but the idea that a man would drop everything, leave his father's house and family business (possibly putting his family in hardship), and follow someone who just walked up and said "follow me" seems pretty miraculous to me.

Well, the Apostle John said:

John 21:25 (KJV)
And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen.

So we know that not everything from Jesus' life was recorded. Luke does record an account of Jesus reading the scriptures in the synagogue, saying it was Jesus' custom to do so (Luke 4:16). So it is possible that Jesus was already known, at least as a godly leader, when he approached these men.

My personal belief is that Jesus already knew each of these men before he called them to leave their professions and follow him. Some think that James and John were Jesus' cousins, which would almost certainly mean that they had know Jesus for most of their lives.

I would point out though, that even if these men knew him their entire lives, it was still a huge step for them to leave everything behind to follow Jesus.