In this passage, Jesus uses language claiming God as a personal father; a begetting father, rather than in an abstract, "God is the Father of Humanity" sort of way.
From here, the Jews performed simple deductive logic; the son of God is a god, therefore Jesus is claiming to be a god.
More specifically, Jesus was claiming to be as divine as God, and therefore endowed with the same authority and doing the same kinds of works.
God the Father doesn't rest on the Sabbath and therefore neither does his son and only God can heal, and Jesus performed a healing, etc
The issue at hand for the Jewish leadership was that Jesus was widely perceived by the people as a prophet; a sentiment which was backed by his continuous miracle working and moral teaching. However, the common understanding was that God does not hear sinners:
Now we know that God does not hear sinners; but if anyone is a worshiper of God and does
His will, He hears him. - John 9:31, New King James Version
In healing on the Sabbath, Jesus was performing a forbidden "work" for which the Jewish Leaders sought to rebuke Him. (See John 5:16-17)
Worse still, in commanding the man to carry his bed upon being healed, Jesus was causing another man to sin. In the perception of the Pharisees, Jesus was guilty of at least 2 sins.
How then could Jesus perform these works if he was a sinner, they wanted to know? In an effort to discredit Him, the Jewish Elders came forward with their accusation and questioned Him.
On the whole, this passage is interpreted by commentators as being a claim of Divinity by Jesus Christ; either by virtue of doing a "divine work," or by his claim to being the Son of God or by virtue of a combination of these factors.
The traditional Patristic interpretation of this passage sees this as Jesus revealing himself to be a partaker of the divine essence, sharing in the power and authority that accompanies it. Basically, if God is not human, but divine, his son must be divine as well.
"Were He not the Son by nature, and of the same substance, this defense would be worse than the former accusation made. For no prefect could clear Himself from a transgression of the king's law, by urging that the king broke it also. But, on the supposition of the Son's equality to the Father, the defense is valid. It then follows, that as the Father worked on the Sabbath without doing wrong: the Son could do so likewise." - St. John Chrysostom, quoted in Thomas Aquinas' Catena Aurea (The Golden Chain)
In this quotation, St. John Chrysostom is suggesting that it would not be a strong enough defense for Jesus to simply say "God doesn't rest on the Sabbath, so why should I?" The King as a prerogative and authority that transcends the rights of his subjects. However if Jesus was equal in privilege to by being of the same nature as God, He would not be guilty of an offense.
What St. John Chrysostom implies in his exegesis of the passage, St. Augustine makes more explicit:
"The Jews however did not understand from our Lord that he was the Son of God, but only that He was equal with God; though Christ gave this as the result of His being the Son of God. It is from not seeing this, while they saw at the same time that equality was asserted, that they charged Him with making Himself equal with God: the truth being, that He did not make Himself equal, but the Father had begotten Him equal." - St. Augustine of Hippo, quoted in Thomas Aquinas' Catena Aurea (The Golden Chain)
Being the begotten Son of God, Jesus shares in the divine nature so is "equal" by nature, entitled to the same privilege and thus exempted from the regulations of the Sabbath.
This same position is held by more recent (relatively speaking) Protestant Commentators:
"rightly gathering this to be His meaning, not from the mere words "My Father," but from His claim of right to act as His Father did in the like high sphere, and by the same law of ceaseless activity in that sphere. And as, instead of instantly disclaiming any such meaning—as He must have done if it was false—He positively sets His seal to it in the following verses, merely explaining how consistent such claim was with the prerogatives of His Father, it is beyond all doubt that we have here an assumption of peculiar personal Sonship, or participation in the Father's essential nature." - Commentary Critical and Explanatory on the Whole Bible by Robert Jamieson, A.R. Fausset, and David Brown
In addition to affirming that Jesus is claiming to participate in the divine nature, notice that the commentators call attention to the "peculiar personal Sonship" which Jesus is claiming as evidence for this participation. In this, the commentators are calling attention to the fact that the language Jesus used to refer to God as His father was much the same as a human would use to refer to their own biological father; this was not an abstract statement of "God as father of humanity" but rather a claim of divine descent.
The commentators also draw attention to the fact that due to His Father's example, and His participation in the divine nature Jesus was exempt from the Sabbath regulations, and like His Father would to continue working.
The sentiment is shared by both John Calvin, and Patristic Authors:
"But it is chiefly concerning himself that Christ speaks, to whom the Jews were more hostile. He declares that the soundness of body which he has restored to the diseased man is a demonstration of his divine power. He asserts that he is the Son of God, and that he acts in the same manner as his Father." - John Calvin, Commentary on John - Volume 1
John Calvin returns to this theme more than once in his commentary on this passage, that the act of healing is a divine work, and that the Son of God is divine, therefore Jesus as the divine Son of God was and authority beyond the regulations of the Sabbath.
"not in the secondary sense in which it is true of all of us, but as implying equality. For we all of us say to God, Our Father, Which art in heaven. And the Jews say, You are our Father. They were not angry then because He called God His Father, but because He called Him so in a sense different from men." - St. Augustine of Hippo, quoted in Thomas Aquinas' Catena Aurea (The Golden Chain)
In this passage, St. Augustine shows his interpretation of Jesus' choice of language as being a demonstration of a "peculiar personal sonship."
"The words, My Father works hitherto, and I work, suppose Him to be equal to the Father. This being understood, it followed from the Father's working, that the Son worked: inasmuch as the Father does nothing without the Son." - St. Augustine of Hippo, quoted in Thomas Aquinas' Catena Aurea (The Golden Chain)
In this passage we see St. Augustine referencing the divine act of healing as another claim to equality of divinity and therefore of authority regarding the Sabbath regulations.
"Christ defended His disciples, by putting forward the example of their fellow-servant David: but He defends Himself by a reference to the Father. We may observe too that He does not defend Himself as man, nor yet purely as God, but sometimes as one, sometimes as the other; wishing both to be believed, both the dispensation of His humiliation, and the dignity of His Godhead; wherefore He shows His equality to the Father, both by calling Him His Father emphatically. (My Father), and by declaring that He does the same things, that the Father does, (And I work). Therefore, it follows, the Jews sought the more to kill Him, because he not only had broken the sabbath, but said also that God was His Father." - St. John Chrysostom, quoted in Thomas Aquinas' Catena Aurea (The Golden Chain)
And finally St. John Chrysostom touches on both themes in the above single passage.