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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