So, let's take this paragraph by paragraph.
I swear by Apollo the physician, and Asclepius, and Hygieia and Panacea and all the gods and goddesses as my witnesses, that, according to my ability and judgement, I will keep this Oath and this contract:
This is basically a more elaborate way of saying, "I swear to God that:". Obviously, the gods mentioned are some of the relevant ones worshiped by the Greeks at the time this oath was penned. I've already introduced these colourful characters in my post here.
To hold him who taught me this art equally dear to me as my parents, to be a partner in life with him, and to fulfill his needs when required; to look upon his offspring as equals to my own siblings, and to teach them this art, if they shall wish to learn it, without fee or contract; and that by the set rules, lectures, and every other mode of instruction, I will impart a knowledge of the art to my own sons, and those of my teachers, and to students bound by this contract and having sworn this Oath to the law of medicine, but to no others.
The second paragraph deals with the social contracts that are implied by the teaching of medicine. In Hippocrates' view, the practitioner of medicine should treat his own medical teacher, together with both sets of sons, as one large and very close family. Of central importance within this tight nucleus is the dissemination of the medical knowledge. One gets the impression that Hippocrates saw medical know-how as a hugely important and powerful tool, to be spread only with those who were worthy. This small exclusive group would consist of one's sons, one's teachers' sons, and other students who had taken the Hippocratic Oath.
I will use those dietary regimens which will benefit my patients according to my greatest ability and judgement, and I will do no harm or injustice to them.
The third paragraph is the only place in the Oath where a specifically recommended treatment is mentioned, namely "dietary regimens". Uncontroversially, it recommends doing your best in this regard, but the latter half of the paragraph ("I will do no harm or injustice to them") has been invoked by both proponents and opponents of euthanasia. It is broad and vague enough to provide ammunition for both camps.
We'll conclude the analysis of the Oath tomorrow.