Hamlet says the above lines to Horatio, a close friend of Hamlet's who studied with him at Wittenberg. Hamlet is explaining to Horatio how he happened to manage to return to Denmark from England.

Prior to these lines, Hamlet explains that he was actually doomed to die in England. He accounts the story of how, while in the middle of the night at sea, he left his cabin in search of Rosencrantz and Guildinstern and found in their things a letter written by the king ordering Hamlet's death upon arrival in England, for the safety of both Denmark and England. He further accounts that he wrote a second letter commissioning the deaths of both Rosencrantz and Guildenstern instead and put the new letter in the old letter's place.

Most importantly in this passage, Hamlet is claiming that Heaven must have had a hand in protecting him because it was due to a feeling of uneasiness that made him search Rosencrantz' and Guildenstern's things. In the lines in question, he is explaining the next things that happened to help him escape his death in England and states that Heaven must have also had a hand in those happenings as well, as we see in his line, "Why, even in that was heaven ordinant" (V.ii.52). The word "ordinant" can be translated as "control," showing us that Hamlet believed God or fate to be in control of Hamlet's life at that moment (eNotes).

Hamlet explains that God must have had a hand in the fact that he was even able to make the letter look authentic because he had his father's signet ring in his purse, which looked very much like the current king's seal, as we see in the phrase "which was the model of that Danish seal" (54). Signet rings depict the official coat of arms for the kingdom and are used as an official seal on documents. Since his father's seal was so much like the king's seal, he was able to seal the envelope and make the switch without anyone noticing at all, thus escaping into safety, as we see in the lines:

...Folded the writ up in the form of the other,
Subsribed it, gave't the impression, placed it safely,
The changeling never known. (55-57)