Meaning: To separate or break apart
To break up is synonymous with the similarly-worded but more literal “break apart”. It plainly comes from “break” + “up”. It originally just meant to break something into pieces, specifically farmland. One can picture plowing a field of hard ground and and the blade breaking it into smaller, softer pieces which are now turned up onto the ground next to it. Similar imagery would be smashing a tile. Broken apart, the tile pieces no longer fit together neatly on the floor—it’s now broken up on other pieces of tile.
This then began to be applied to groups of people in the 1700s. If a group of people was protesting or fighting or gawking or just generally being a disturbance, it would often be “broken up”. The people would be separated and dispersed as if it had been a solid object.
It is also at this time that it began to apply to the most common use today: relationships. Like a crowd of people could be broken up into individuals and sent away, so two people can be broken up from a couple back to a single man and a single woman. You break the relationship and separate the people. “Breaking up” is the most common way of saying that a romantic relationship has ended.
Eventually, this also became the language used for telecommunications. When you talk to someone on the phone, there can be a break in the connection, separating the speakers. When the phone call is beginning to degrade and intermittently losing connection, it said that you are breaking up. This is especially common in cell phone networks, where reception can be spotty, and in VoIP calls, where packet loss often leads to only bits of conversation getting through.
Ultimately, “break up” has quite a few distinct usages, but they all share essentially the same meaning of breaking something apart so that it is separate.