What was the reason Jesus allowed Christians to eat pork?

Jews and Pigs

We don't really know.

The problem with this one is that lots of people have plausible-sounding reasons for the prohibition. Lots of people have implausible-sounding reasons for this too. Beyond, "because the Torah says so", none of them are really provable, or have any kind of serious historical consensus. Even within the Torah, Genesis 9:3 implies its just fine to eat them, so it isn't consistent with itself on the subject.

Every moving thing that lives shall be food for you; and just as I gave you the green plants, I give you everything.

(NRSV)

It should be noted that Deuteronomy 14:8, which explicitly forbids eating pig meat, is today considered by textual analysts to have been authored from a different source than the Genesis passage above. However, The Deuteronomist is likely an older source than The Priestly Source responsible for that bit of Genesis (by a couple of centuries), and there's little indication that pork consumption was briefly acceptable during that time.

Mark 7:19

Mark was written down most likely sometime around 66–74 AD. While that's more than a generation after the death of Jesus, it would still make it the oldest Gospel (so in some ways our best historical source for Jesus).

This was right around the time of the destruction of the second temple. Prior to then there was a faction of the new sect that believed in keeping it a Jewish-only religion. This faction was led by figures such as James and Peter and was headquartered in Jerusalem. Another faction believed in opening the new faith up to the world ('Gentiles'), and was led by many missionaries throughout the Roman world, most famously Paul.

It was this latter faction that wrote Paul's original epistles, and then later Mark. This faction clearly had a vested interest in converting Gentiles, so you find a lot of things in their writings that are convenient for that. Specifically the removal of the requirements for Jewish conversion (and particularly circumcision!), and not needing to follow the strict Jewish cleanliness and dietary restrictions. This is almost certainly why you find that passage in Mark. Notice how the implication is then helpfully spelled out explicitly. That very rarely happens in the Gospels. Whoever wrote that didn't want any argument whatsoever on the matter. Paul's epistles (which remember were written earlier than Mark) indeed do hint that there was a great deal of argument on this issue in the early Church. What better way to end the argument, than to put it in Jesus' mouth?

About the time Mark was being written, the Romans got entirely fed up with the constant Jewish rebellions and occasional destruction of Roman legions. Rome laid siege to Jerusalem, burned it to the ground, and killed or enslaved and deported everyone living there. This also had the effect of decapitating (if not completely wiping out) the "Jewish-only" Christian faction. If they had their own gospels putting other words in Jesus' mouth, they burned with the city.

So basically what happened is that this pro-Gentile passage was useful to have for one of the two early Christian factions, and Gentiles (Romans) ended up long-term being the right horse to bet on. So that one early Gospel survived to this day.

The Gospel of Matthew however, appears to have come later from a Jewish Christian community that survived the destruction of the second temple. It was much more concerned with preserving the Jewish nature of the religion, and to that end used Jesus' voice to directly contradict Mark on this matter in 5:17-18.

Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished.