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I'm no expert on the applicable law regarding tariffs and international trade, but I do know that the most famous tariff policy in US history was the Smoot-Hawley Tariff Act, named after two members of Congress who were its principal sponsors. It added various new tariffs to US law, and is generally considered a disaster by historians because of the damage that the retaliation it prompted did to the US economy. The notable thing, for the purposes of this question, is that it was a law: a bill originating in Congress, passed by legislative process, and sent to the President to sign.

More recently, during the Trump administration, we've heard a lot about the administration imposing new tariffs as a bargaining chip to use against China, and then relaxing them as negotiations with China go well. News reports generally speak of the President imposing the tariffs, which can be interpreted either personally or as a synecdoche representing his administration, but either way it's referring to Executive Branch action.

How does this work, when tariffs, by historical precedent, are a matter of law and the responsibility of the Legislative Branch? Are the common news reports oversimplifying things, or has some action been taken to delegate this responsibility to the Executive for some reason?

Congress granted to the President authority to set tariffs by agreement with outher countries in the Reciprocal Tariff Act of 1934, and later extended such authority under various laws. The trump tariff changes were imposed under Section 232 of the Trade Expansion Act of 1962, as described in "America Trades Down" from Lawfare The article suggests that there may be challenges to these tariffs. It seems that Section 232 has been rarely used, and not since the creation of the WTO in 1995.

The president can order that tariffs be imposed for national security. Trump's reasoning is laid out here:
https://www.vox.com/2018/3/8/17097206/trump-tariffs-congress

You are right that the constitution gives congress the right to establish tariffs, but congress has ceded some of that power to the Executive Branch of the government over the years, but only for national security purposes or during an emergency.

The act that Trump's administration used is summarized here:

To conduct an investigation under Section 232 of the Trade Expansion Act of 1962, the Secretary of Commerce may self-initiate the investigation or an interested party may initiate an investigation through an application. Any investigation initiated must be reported to the Secretary of Defense which can also be consulted for information and advice should any policy questions arise during the investigation. The Department of Commerce reports its findings to the President within 270 days of initiating any investigation, with emphasis on whether certain imports threaten to impair the country's national security. The President has 90 days to formally concur or not with the report received from the Commerce department. If s/he concurs, his or her statutory authority under Section 232 allows him or her to modify or adjust the imports as necessary though tariffs or quotas. In effect, following the report submitted, the President of the country may take a range of actions, or no action, based on the Secretary's recommendations provided in the reports.
https://www.investopedia.com/terms/s/section-232-trade-expansion-act.asp

The other answers are only partially correct. Not all Trump tariffs are based on the national security section 232. In fact, most tariffs he raised on China are not like that, but rather:

On March 22, 2018, Trump signed a memorandum under the Section 301 of the Trade Act of 1974, instructing the United States Trade Representative (USTR) to apply tariffs of $50 billion on Chinese goods. Trump stated that the tariffs would be imposed due to Chinese theft of U.S intellectual property.

Actually the complete wording of section 301 investigation was

“China’s unfair trade practices related to the forced transfer of American technology and intellectual property.”

The latter section:

authorizes the President to take all appropriate action, including retaliation, to obtain the removal of any act, policy, or practice of a foreign government that violates an international trade agreement or is unjustified, unreasonable, or discriminatory, and that burdens or restricts U.S. commerce.

The later $200 billion raise was also based on section 301, but basically as counter-retaliation to China's response to the initial tarrifs:

After separate notice and comment proceedings, in June and August USTR released two lists of Chinese imports, with a combined annual trade value of approximately $50 billion, with the goal of obtaining the elimination of China’s harmful acts, policies and practices. Unfortunately, China has been unwilling to change its policies involving the unfair acquisition of U.S. technology and intellectual property. Instead, China responded to the United States’ tariff action by taking further steps to harm U.S. workers and businesses. In these circumstances, the President has directed the U.S. Trade Representative to increase the level of trade covered by the additional duties in order to obtain elimination of China’s unfair policies. The Administration will continue to encourage China to allow for fair trade with the United States.

And unlike section 232, section 301 has been invoked more often:

Since 1974, the USTR has initiated 125 Section 301
cases, retaliating in 17 instances.

but most of those retaliations were before the Uruguay Round of the WTO:

After the United States implemented the UR [Uruguay Round] agreements
and joined the WTO is 1995, the USTR still sometimes
began Section 301 investigations but then brought the
issues at hand to the WTO for dispute resolution. After
2010, the USTR brought all trade disputes involving WTO
members directly to the WTO for adjudication. The Trump
Administration’s use of Section 301, rather than solely
utilizing the WTO dispute settlement process to address the
issues of concern, is a departure from past U.S. practices.

Prior to the UR agreements, China was a major target of
Section 301 actions. In 1992 and 1994, the United States
threatened to impose increased tariffs against China over its
IPR [intellectual property] policies. In 1992, the United States threatened
increased tariffs on $3.9 billion worth of Chinese goods
over market access issues. These cases resulted in bilateral
agreements before tariff hikes were implemented. In
October 2010, the USTR launched a Section 301
investigation into Chinese policies affecting trade and
investment in green technologies, and in December 2010,
brought a WTO dispute settlement case against China, but
only in regard to its wind power subsidies. In March 2012,
the USTR initiated a WTO dispute case against China’s
export restrictions on rare earth elements (used in a number
of green technology products). The United States largely
prevailed in both cases.

As for a bit of historical background, the 1974 Trade Act was passed in the context of the Tokyo Round of GATT, and on the background of stagflation in the US, coupled with the 1973 OPEC oil embargo (itself related to Yom Kippur War).

Later on, Congress tried to put some limits on the retaliatory power delegated in 1974 via section 301 (but also granted him foreign-investment stopping powers):

Amendments to section 301 in 1979 established specific time lines for investigations and the final resolution of disputes. Although these amendments created a more elaborate regulatory framework for dispute settlement, they did not deprive the President of discretion regarding whether to take action. The amendments also required consultations with the government named in the petition.

More amendments in 1984 defined "unjustifiable," unreasonable," and "discriminatory practices" and provided for the initiation of section 301 investigations by the USTR. The law required the preparation of an annual National Trade Estimate or NTE and
permitted the President to place restrictions on foreign direct investment.

The most recent amendments, in the Omnibus Trade and Competitiveness Act of 1988, transferred final decision-making authority in section 301 cases from the President to the USTR.

And so goes the theory that the USTR is somewhat independent from the President. Which I think is largely ignored by the press today. Probably the fact that Trump tweets every time that the USTR does something in relation to tariffs is undoubtedly related though.