The origins of the phrase have been disputed. What's clear as far as I can tell is that the phrase, generally, has been associated with those who believed that a Jewish state had no right to exist, and that it arose in the 20th century during the dispute between Zionist Jews seeking statehood and Palestinian Arabs seeking statehood, each with different visions of how that should look and whether the other should have a similar claim. There have long been references to "the sea" in slogans relating to the conflict, some disputed, such as claims that Arab leaders called to "push Jews into the sea" during wars in 1948 or 1967.

The slogan itself has a much richer history (of claims) post-establishment of the PLO, but even so, I haven't been able to verify many of the claims. Some say that it was popularized by the PLO, but I can't verify that either. What is clear is that it has been associated with the more extreme positions in the conflict. With the founding of Hamas in 1988, the slogan became even more popular, and featured prominently in rallies and marches. Matthew Levitt and Dennis Ross mention it in Hamas, describing a convention of the Islamic Association for Palestine in the US (a group that a US court ruled aided and abetted Hamas). They describe a convention held in 1989 (and in honor of Abdallah Azzam, bin Laden's spiritual mentor, who had recently been killed by a rival Islamist group), where a banner at the front of the room "proclaimed: 'Islamic Palestine from the river to the sea' in Arabic, while another banner bore Hamas' name in Arabic." Mishal and Sela, in Palestinian Hamas, likewise quote Hamas use of this phrase directly, and note that after an internal debate over whether to potentially engage in the peace process the PLO was moving towards, the reaffirmation of "from the river to the sea" by Hamas was used to continue drawing a contrast between the PLO and Hamas.

Palestinians in the PLO have also acknowledged the slogan's general meaning, which is the removal of Israel. Ahmed Qurei, a prominent Palestinian negotiator, wrote in Beyond Oslo that Israeli demands that the PLO amend its Charter (which called for Israel's eventual removal, though not in the slogan's terms directly) were likely to backfire, saying, "Israel's very insistence on seeing it amended had led some of us to feel that perhaps after all it should be retained as part of our historic goal to liberate Palestine 'from the river to the sea'." This acknowledgment of the historic goal being, as such, the end of Israel as a state, should give you the gist.

So we can say that the idea has historically grown from slogans frequently invoked to deny one or the other group statehood, and is more often associated historically with the positions of groups like Hamas or the Palestinian goal broadly of removing Israel as a state. What follows that removal is a point of difference in interpretation. Hamas, of course, would like the implementation of an Islamic state, and its charter and subsequent policy documents and statements all reinforce this. That's not to say that it would be similar to ISIS in practice, but Hamas has historically called for a far more religious-Islamist policy than the PLO, by comparison, and would like such a state to replace Israel.

The PLO, for its part, has claimed that such a state would be secular in nature, and sometimes has claimed it would be democratic. However, it also has in the past said only (per its charter) Jews who had "normally resided in Palestine until the beginning of the Zionist invasion will be considered Palestinians". That would discount a huge swath of Israeli Jews today, and would mean they were not Palestinian. In the past, that might've led to rhetoric about them being expelled, and comparisons to the French Algerian historical record. However, others argue that this state would simply become "binational" and thus a single, democratic state.

Notably, however, this has also been rejected by Israeli Jews, for the same reason a majority-Jewish democratic state has often been rejected by the PLO (and why the "quiet part" of the call for a single state by Palestinians often included removing many Jews viewed as "settlers" for having arrived post-1917). The view held by most Israelis was expressed by the controversial Vladimir Jabotinsky, a Revisionist Zionist leader (the far less popular movement pre-1948, who founded the Irgun militia), who said that such a solution would simply create a 23rd, 24th, 25th Arab state while removing any chance for the first and only Jewish one. In short, the Israeli position is that Jews have not had statehood or self-determination in true form (a right guaranteed in international law) in centuries, and have a right to that self-determination. Removing the only Jewish state where Jews have the ability to self-determine to replace it with a majority-Arab democratic state would be removing said right, and handing it to the new ethnic majority. This, of course, is the same argument used by Palestinians who wanted a two-state solution when it was unclear that there would be an Arab majority in a one-state solution (even today, the clarity on who holds the majority is often overstated). You can even find a video of Benjamin Netanyahu, just 28 years old at the time (an economic consultant back then, and a graduate of MIT) going by "Benjamin Nitay", debating the conflict. The debate opens with a question about whether the core of the conflict is self-determination. He says no, and the core is (exchange below):

"The unfortunate Arab refusal to accept the state of Israel. And I think as we mentioned earlier, for 20 years, the Arabs had both the West Bank and the Gaza strip, and if self-determination were, as they now say, the core of the conflict, they could have easily established a Palestinian state then, but they didn't."

"When did the issue arise then?"

"Well for 20 years, we didn't hear a word about self-determination. And in fact, what we did hear, those of us living in the Middle East, was about driving the Jews into the sea."

He attributes the self-determination argument to "dressing up" the slogan to remove Israel. Then the moderator asks Mr. "Nitay":

"Do the Palestinians have a right to a separate state, Mr. Ajjami [the other debater, name uncertain to me] has been talking about human rights?"

"Well, I don't think they do, but I think that it's quite instructive that the Palestinians who are invoking the right of self-determination, which is an attribute for separate nations, themselves are the ones who define themselves as part of the Arab nations. No one is denying that there are Palestinian Arabs, there is a very distinguished Palestinian Arab sitting right next to me. But the Palestinians themselves, in the Palestinian National Covenant, the very first article, say that the people of Palestine, quote, are part of the Arab nation. Well let's look at the Arab nation. It has 21 states, an area roughly the size of the United States, and one sixth of the entire world's wealth. Now add to that, the fact that there already exists a Palestinian state, and that is Jordan, 60% of whose population is Palestinian...so what we're talking about is for a 22nd Arab state and 2nd Arab state."

"What should be done with the Palestinians in the West Bank, it's a problem, so what should be done, in your opinion?"

"Well I think that the Palestinians in the West Bank are going to be offered the full human rights, the full civil rights, as no Arabs are offered in the Middle East...I'm all in favor of having the same Arabs living in the West Bank and the Gaza Strip being offered [the right to vote] in the final peace agreement."

You'll notice, of course, that this would mean Israel annexing the West Bank and Gaza. And this was said, of course, at a time of Jewish population majority in the region. The PLO, naturally, viewed such a one state solution as a negative outcome, and therefore pushed forward the argument of self-determination. Now, Netanyahu's arguments are distinct from how Israeli views developed; some of his views are representative of Israeli popular opinion at the time, some are not.

Continued in part 2 in a comment below.