Now that the Jewish leaders know that Jesus heals on the Sabbath, they start to not like him because they think Jesus breaks the rules. Thus, persecution begins, and in response, Jesus proclaims how "[His] Father is always at work" and how "[He] too [is] working" (5:17). This brings up another reason why Sabbath healings are justified (aside from how carrying the bed is technically not part of a job). The Sabbath is meant for people to focus on God, which is why God works and also why humans don't do their job. Jesus reveals the character of God through healings.
But then the Jewish leaders raised another question -- Jesus claims God is his Father, which makes him equal to God (5:18). In response, Jesus explains that the Father and Son work together. For instance, the Son "can do nothing of Himself" but does acts of the Father in like manner (5:19). In essence, the Father and Son have similar -- if not the same -- character (like Father, like Son). Additionally, the Father "loves the Son" and "shows Him [everything the Father does]" and "will show Him greater works" (5:20). The Son has a strong bond with the Father and follows the Father in what the Father does. Later, the Son will give examples of the relationship in action.
This section conveys the idea that Jesus is God. I know that the "Jesus is God" thing may be a bit confusing, but for now, think of it as how both have the exact same character (which may explain the bond and the following).