It can be summed up by 3 things (and if I missed any, feel free to point out).
The success of Reagan and the re-aligning of the party behind his ideals.
Conservative talk radio and the ending of the fairness doctrine.
The 1976 Republican Convention was a borderline disaster. Think of Bernie's effect on Hillary's convention victory and multiply that by 100.
Gerald Ford came into the convention the clear favorite, with more delegates and more votes but without the number of delegates required to secure the nomination (Similar to Donald Trump, but in Trump's case, the other candidates bowed out and left Trump a clear path to victory). In 1976, Reagan had a loyal following and he thought he could challenge and win over some delegates and pull out a victory. He received a 10 minute standing ovation at the convention and his concession speech left many thinking "We voted for the wrong guy". What's more, the Reagan supporters LOVED Reagan. The Ford supporters, mostly liked Ford, and while Ford had more numbers behind him, the Reagan support was incredibly vocal and could be seen as a galvanizing of true conservatives.
If you go back to Richard Nixon, Nixon felt that the heart of the presidency was foreign policy. That sounds crazy today, it was a more common view in from post WW2 into the 1970s. Nixon didn't care as much for domestic policy and he didn't lean all that conservative on it. He founded the EPA for example.
The Republican party in the 1970s had conservative wings, the anti abortion wing for example and a few holdouts from the Dixiecrats, but it was nothing like the Republican party today where tax cuts and anti immigration are mandatory policies. Jimmy Carter was arguably the spending cut candidate in both 1976 and 1980, not Ford or Reagan, though Reagan did cut some spending in places, he increased it in others.
But I digress. I'm mostly trying to point out how the parties were different pre-Reagan to post-Reagan. Reagan's tax cuts, in 1980 were uncertain and many republicans had concerns and questions about them. George HW Bush called Reagan's economic plan "Voodoo Economics".
By 1988, Reagan's tax cut approach was seen as a great success and pretty much the entire Republican party aligned behind it. Now, tax cuts aren't all there is to a Republican congress moving farther right, but it was part of it. No good Republican supported tax hikes after 1988. George H.W. Bush got egg on his face for his Read my lips promise and then, 3 years later, raising taxes to pay for the gulf war.
Conservative talk radio
While the conservative movement (as I said above) was galvanized in 1976 and strengthened behind Reagan's success, the influence of conservative talk radio shouldn't be dismissed. It was (and still is) very popular and it reached millions of listeners. Loudmouths like Rush Limbaugh were telling people that conservative is good, liberal is bad, and people listened. It's impossible to say how big an effect radio had, but it was a loud voice that had millions of listeners. It was free advertising and free support for any politician who was far-right enough. Moderate Republicans could even find themselves the target of far-right radio.
Radio also made voting far right feel like a populist movement, when it was also, at the same time, very much a political power movement, but political power also depends on being able to win elections.
While it's often overlooked, the influence of Party Primaries is key to the republican party moving further right. "My opponent isn't conservative enough" and well funded conservative candidates are often able to win primary elections. There are exceptions where the moderate appeals to the masses and wins, but on average, America's political division means that the republican primary often leans right and the democratic primary (to a lesser extent), leans left.
A political candidate really only needs 25% of the vote behind the to win. If half the voters are left and half are right, and half the right voters like the other candidate, but the other 75% do not, 25% popularity can win a primary and then a 50/50 split along party lines can win the office.
Take Mitt Romney for example. (though he's not a great example because he lost), but when he ran for president he moved to the right so as to secure the victory in the republican primary. He was never that far right before, but he moved in that direction to win the primary. It's become necessary for many Republicans to show how far right they are to win primary elections and that's half the battle, and in Red states, it's 90% of the battle. Win the primary, the Republican is 90% guaranteed to win the office in deep red states.
This was less of an issue before the parties were so divided, but with the clear division in the party today, the primary system lends itself towards extremes, and this is especially true on the right where "My opponent's not a real conservative" can hurt the opponent and win an election. Republicans have to be very careful to not show any left at all today, and that's a lot of why the party has moved further and further right.
There are arguably other factors. PACs and Superpacs, which came more recently and money going to far right conservatives, and political strategists, like Karl Rove who's been called A Mastermind behind the Republican party. But the primary drivers are the success of Reagan, the galvanizing of people behind the Reagan model, and the wide divide between the two parties today and the nature of the primary election, which cuts the vote in half, and the "More true conservative" often has a clear path to office.