There are two different "Title 47"s in play here. Title 47 of the US Code, as referred to by rclocher3, contains laws passed by Congress on the subject of telecommunications. Most interesting for our purposes is Chapter 5, Subchapter 1 which establishes the FCC. It sets the compositon and appointment rules of the FCC, regulates how the FCC can spend money and how it can obtain revenue, and most importantly, in 47 USC 154(i), says that
The Commission may perform any and all acts, make such rules and regulations, and issue such orders, not inconsistent with this chapter, as may be necessary in the execution of its functions.
Which basically incorporates any rules the FCC makes as part of the law, given that they are within the purpose of the FCC as defined in the chapter preamble, and they don't conflict with the remainder of the law.
Those rules are cataloged and collected in Title 47 of the Code of Federal Regulations, which is what you linked to in your question. These are not established by acts of Congress, they're enabled by the act of congress that gives rulemaking power to the FCC. All of Part 97 is contained within 47 CFR. To change it, the FCC only needs to follow its own rules about how regulations are made. So you're correct about how the delegation works, and with few exceptions, it doesn't take an act of Congress. However, Congress can act by either passing a law mandating that the FCC takes a certain action, or by passing a law which directly overrides the FCC rules.