The phrase originated probably from other metaphors related to surfing that developed in the 1970s/1980s. One example is channel surfing. These metaphors reflected jumping from place to place in order to find something. These mentions were encoded in print in the 1980s, and possibly before (https://www.phrases.org.uk/meanings/surfing.html gives some good detail).

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The part I wanted to add to this question I thought more interesting is how surfing the web reflected the use of metaphors in general. There is a paper by Jamet that discusses the use of metaphor in the development of the internet that goes into much more detail about some of the reasons that the "surf the web" phrase and discusses it in other languages, particularly French. The main takeaways are as follows:

1. The internet is vast, much like the ocean, so ocean metaphors are very appropriate. One example of this is Netscape, the early browser which used a nautical steering wheel in early designs
2. We must navigate the space of the vast internet, which again goes to the same metaphors (Netscape navigator) is another example
3. This navigation was mostly a sense of pleasure or adventure, rather than duty. Surfing is an activity of enjoyment and free choice. This contrasts with other early metaphors like "the information superhighway"
4. Much of the early exploration of the internet was from jumping from link to link, which again relates to early surfing navigation.

About the decline, I can't add sources, but here are my guesses:

1. Each generation likes to add its own words to speaking about the internet. McCullogh's Because Internet is a good recent book to summarize this. Using the old language is stigmatized by each "generation"
2. More of the internet is curated and not from jumping from link to link (which Jamet speculates)
3. The word "web" is now especially old-fashioned.

Citation:

Jamet, D. L., & Moulin–Lyon, J. (2010). What do Internet metaphors reveal about the perception of the Internet. Metaphorik. de, 18(2), 17-32.