Really good to see that you are reading John's gospel. I hope it's not too heavy going. Here's the background to the verse you asked about
Now in Jerusalem by the Sheep Gate there is a pool, called in Hebrew Beth-zatha, which has five porticoes. In these lay many invalids—blind, lame, and paralysed. One man was there who had been ill for thirty-eight years. When Jesus saw him lying there and knew that he had been there a long time, he said to him, ‘Do you want to be made well?’ The sick man answered him, ‘Sir, I have no one to put me into the pool when the water is stirred up; and while I am making my way, someone else steps down ahead of me.’ Jesus said to him, ‘Stand up, take your mat and walk.’ At once the man was made well, and he took up his mat and began to walk.
Now that day was a sabbath. So the Jews said to the man who had been cured, ‘It is the sabbath; it is not lawful for you to carry your mat.’ But he answered them, ‘The man who made me well said to me, “Take up your mat and walk.”’ They asked him, ‘Who is the man who said to you, “Take it up and walk”?’ Now the man who had been healed did not know who it was, for Jesus had disappeared in the crowd that was there. Later Jesus found him in the temple and said to him, ‘See, you have been made well! Do not sin any more, so that nothing worse happens to you.’ The man went away and told the Jews that it was Jesus who had made him well. Therefore the Jews started persecuting Jesus, because he was doing such things on the sabbath. But Jesus answered them, ‘My Father is still working, and I also am working.’ For this reason the Jews were seeking all the more to kill him, because he was not only breaking the sabbath, but was also calling God his own Father, thereby making himself equal to God.
Jesus said to them, ‘Very truly, I tell you, the Son can do nothing on his own, but only what he sees the Father doing; for whatever the Father does, the Son does likewise. The Father loves the Son and shows him all that he himself is doing; and he will show him greater works than these, so that you will be astonished. Indeed, just as the Father raises the dead and gives them life, so also the Son gives life to whomsoever he wishes.
In the gospel of John, "Jews" often means "Pharisees". The Pharisees were a powerful group at the time and they placed great value on private religious observance, and had a very strict interpretation of the Law. The Ten Commandments says "Remember the sabbath day, and keep it holy. Six days you shall labour and do all your work. But the seventh day is a sabbath to the Lord your God; you shall not do any work". The Pharisees introduced many additional rules to clarify exactly what could and could not be done on the sabbath. One of their rules was that it was not permissible to carry anything - and that's where the quarrel begins in this story. Jesus told the man to pick up his mat, and that fell foul of their rules.
Jesus - to the exasperation of the Pharisees - responds by saying that the Father is at work, and so he is at work too.
In the flow of the argument, "the Son can do nothing on his own" does not mean that Jesus is saying, "I am useless without the Father", he is saying "Everything you see me doing is what the Father is doing".
Does that help explain it? Tell me if not, I'll gladly say more if it will help.
