Great question.
From 1932 to 1994, the Democrats controlled the House of Representatives except for two short periods because of, well, the Depression and the South. There are a lot of other factors like race, gerrymandering and incumbency effects, but economics was a major driver of Southern voters staying Democrat, and keeping the Democrats in power in the House.
But first, we need to talk about party systems. These are epochs in which elections are subject to a certain dynamic, and the dynamic changes in each system. Seymour Martin Lipset probably has the most accessible volume on this: Party Systems and Voter Alignments. The US is believed to have had at least five party systems, probably six, maybe more.
There is a LOT of debate in the literature about when the Fifth Party System ended, and if it ended, etc. But the key thing is that the Fifth Party System was one of Democrat dominance, and that Democrat dominance extended - especially in the House elections - into the Sixth Party System, right up to 1994.
The third and fourth party systems were a function of the Civil War. Basically, White Southerners voted Democrat in the "Solid South." There just weren’t white Republicans in large parts of the South (and blacks were prevented from voting by force.) Obviously, the biggest reason for this was the Civil War and Reconstruction. Northern Democrats were often immigrants voting to support a patronage machine that provided jobs in exchange for votes. Republicans were the party of professionals, Protestants, and prohibition. As time went on, both parties were split between economic progressives and conservatives. Where blacks could vote, they mostly voted Republican.
The 1932 election was what political scientists call a “realigning election.” That means the previous party system ends and a new one begins. 1932 is the most dramatic one in US history. From 1865 to 1932, federal politics in the US is mostly dominated by the Republicans. From 1932 to at least 1994, federal politics in the US is mostly dominated by the Democrats. This is because as a watershed in US election history, the Great Depression stands second only to the Civil War.
The Democrats win more than 70% of the seats in Congress in 1932. What happens is that the “Solid South” of Democratic representatives in Congress is suddenly joined by an influx of Democrats from Northern and Western communities. This is the “New Deal Coalition” and it is composed of Northern and Western liberals who believe in government spending joined onto the traditional Democratic coalition of Southern whites and northern city machines. And while the Northern and Western Democrats are mostly liberal, the Southerners are often quite conservative.
The New Deal Coalition holds together through the Second World War due to Roosevelt’s popularity. He is seen to have saved the economy and established key anti-poverty programs like social security that become totems for poor voters, including Southerners. And while the Southern white Congressmen may be racists, they really don’t like Hitler. They back Roosevelt strongly during WWII. And that sets up an important pattern. Democrats can hold the South when talking economics or a strong defense, but they have to be careful about issues of race.
Truman loses the House and Senate in 1947. Truman then runs against this “Do Nothing Congress” in his historic upset in 1948, and wins back Democratic majorities. He does this by emphasizing economic issues and talking about how the Republicans are obstructing that agenda. The economically distressed rally to that call, especially in the South and West. Blacks begin to become a key vote to win the big industrial states, and are increasingly moving to the Democrats across the country.
When a popular Eisenhower wins the Presidency and the House and Senate in 1952, the new Democrat minority leaders are the cagey Southerners Lyndon Johnson and Sam Rayburn. Robert Caro’s Master of the Senate describes how Eisenhower is personally hugely popular, but the Republican Party is split into a moderate faction and a hard-core Conservative faction. Rayburn and Johnson smartly unite the Democrats to actively support the Eisenhower agenda MORE than the Republican leadership. As a result, the House Democrats are basically running for Ike when he is popular, and they recapture thin majorities in both houses. A recession in the late 1950s boosts the Democrats who can say "Republican times are hard times."
In 1964, Johnson gets the Civil Rights Act passed and this sets off a very long fuse on a very big bomb. African Americans were a late addition to the New Deal Coalition that allows Democrats to dominate the northern and Western cities. The Southern racists leave the Democrats for the GOP over Civil Rights. You now see Republicans regularly winning elections in the South, but the average Southern white may also keep voting Democrat at the House level over economic issues. (See Kuziemko and Washington.)
The move from the fifth to sixth party system is ragged and slowest at the House level. The old Depression era argument that “Republican times are hard times” has a lot of power in the South for House voting. Southern Democrats in Congress carefully walk a line on racial issues while emphasizing a strong defense and social security to stay in office. VO Key says the voting history of the South is about race alone, but that doesn’t explain the Democratic hold on the majority of southern House districts into the 1990s. Instead, it is partially race but also very much about economics, according to Johnston and Shafer.
Think about it like this: the Democrats are controlling what they do now in the House PLUS they still have half the South. A voter in Tennessee might vote for Nixon or Reagan, but also vote Democrat for the House and State House and Governor.
Watergate boosts the Democrats in Congress as voters voice their anger at the Republicans. Jimmy Carter’s run in 1976 has the Democrats winning the South at a Presidential level. These two events provide enough of a boost that the Democrats had the majority of southern congressional representatives right up to the 1994 election.
Right through the 1980s, the Democrats also had advantages with redistricting and incumbency. The boundaries of congressional districts were often set by the state legislature, and the state house was often controlled by the Democrats even in the South. It meant a congressional district for a Democrat incumbent could be manipulated to maximize their chance at reelection. In the 1980s, the GOP made it a major strategy to win state legislatures, and thus the control over redistricting.
Incumbents also had huge advantages in name recognition, fundraising, and delivering patronage and projects back to their voters. Southern Democrat representatives were often long serving and thus had best chance of bringing home the bacon. Voting Republican meant getting some rookie who had no power. For lower income Southern districts, a long-time Democratic incumbent was a major source of economic activity; voting him out was a hit to the pocketbook. And those Southern Congressmen were Cold Warriors. They were aligned with conservative voters on defense and that military spending came home as bases and military contracts.
With the GOP gaining control of state legislatures, the gerrymandering effect was reversed. As the Southern economy grew, long-time incumbents bringing home the bacon became less important. The Cold War ended, so fighting the Soviets stopped being a way for a Democrat to prove he's got Southern values.
So when Southern Democrat Bill Clinton was positioned as more interested in issues like gays in the military and "socialist" health care, those Southern incumbents got blown away. In 1994, the “Solid South” that had been with the Democrats since 1865 finally shattered. For most of the period since then, there was not one white Democrat elected to Congress in the Deep South. Not one.
TL,DR The dominance of the House by Democrats from 1932 to 1994 was a function of Southerners staying Democrat because of the Civil War and the Depression and the Southern economy. Civil Rights pushed those Southern whites to the GOP. Incumbency and gerrymandering and the Cold War masked the impact for a while, but eventually, Southern whites moved as a complete block to the Republicans.
Edit: Cleaned this up a bit for readability.
Seymour Martin Lipset, Party Systems and Voter Alignments
Robert Caro, Master of the Senate.
Kuziemko and Washington, https://economics.yale.edu/sites/default/files/files/Faculty/washington/south-dems.pdf
Johnston and Shafer. http://www.hup.harvard.edu/catalog.php?isbn=9780674032491
VO Key, Southern Politics