Federal Power
While the New Deal Era in the United States saw a great expanse in the powers of the Federal Government, and specifically the Executive Branch, it didn't stand alone. There are major milestones that empowered this populist shift.
In response to failed attempts to enforce an income tax on the populist, the 16th Amendment was ratified in 1913, with great populist fervor. This shifted more monies towards the Federal Government. Additionally in this populist wave, the 17th Amendment pass, transforming how Senators were assigned. With the shift to a popular vote instead of appointed by the State Legislatures the designed influence of the States on the Federal was reduced.
Further down the road, in the midst of the Depression, there was the transformative era 4 term run of FDR, with a stronger shift towards Federal Power. Chief among them was FDR's efforts to cajole the Judicial Branch to restrain his efforts, threatened to pack the court with sufficient judges favorable to his cause.
From that mindset, there was the shift in interpretation of the Tax and Spend Clause of the Constitution, leading to the construction of the now well bodied General Welfare Clause. In a minor aside amidst United States v Butler, which was decided against the Government, the Tax and Spend clause was transmogrified into a new, broader enumerated power. The Federal Government was now empowered to spend money as it willed. This great increase in Federal Power doesn't fully explain the growth in executive power.
Executive Power
The checks and balances of our Federal Government were designed on the axiom that individuals are greedy, and that its members would act with their enumerated powers to protect the same while checking against encroachment from the other branches. While a grand theory, it has proven to be a false assertion.
Congress is a cowardly branch; while there may be some members of integrity, by in large they enjoy the power of their office and spend much of their time trying to get re-elected than doing the Job. For that, they've learned over the decades that while doing something grand is a good way to stay in power, it entails a significant level of risk. Instead, they can do nothing offensive, while occasionally shifting funding back to their states via "general welfare" spending, to further enshrine their office. How is the government to run if Congress becomes risk adverse to passing actionable legislation? Through Intelligible Principle of course.
Intelligible Principle, (J. W. Hampton Jr. & Co. v US), is the legal fiction saying:
In a unanimous decision, the Court held that Congress, within "defined limits," could vest discretion in Executive officers to make public regulations and direct the details of statutory execution. The Court argued that the same principle that allowed Congress to fix rates in interstate commerce also enabled it to remit to a rate-making body under the control of the Executive branch.
Now, Congress can pass 'enabling legislation,' delegating great amounts of legislative power to Regulatory Agencies in the Executive Branch. Instead of politicians, answerable to the populace at the ballot box, governmental power becomes vested in bureaucrats of the Executive Branch.