I will address two aspects of Creative-Commons licences seperately:
Open Access and not giving somewhat exclusive rights to a journal
The reason why journals do not give everybody access to their articles and want some exclusive rights on the article is that they need to pay their expenses (typesetters, copy editors, printing, maintaining editorial managemetent system, …) and do so by selling articles and issues (mostly to university libraries and similar). You can change or circumvent this by:
Publishing in an open-access journal. In this case, it’s usually you and not the reader who pays for the journal’s expenses. Since there is no need for the journal to get exclusive rights, you can usually publish the article under a Creative-Commons licence. Note that some funding agencies give you money exclusively for publication costs, which you can spend on this. Be aware though that there are more black than white sheep in this field. Finally, the price of publication is debatable (but so is the price of pay-to-read journals).
Nowadays, many journals offer pay-to-publish open-access options, which are like the above, just that your article is published in a classical journal instead of a pure open-access journal.
Many journal allow the authors to disseminate preprints, e.g., via the ArXive. Though they still get some exclusive rights to the article and there may be some restrictions, there is de facto open access to your article. SHERPA maintains an extensive database on what kind of preprint publication is allowed by which journal.
As to why researchers do not care more about this: The old publication system is an established structure which takes time to change. Theoretically, we could do with one central, globally funded open-access publication system (in particular by avoiding printing costs). Also note that in technologically inclined fields, publishing preprints is very common, so there already is open access to most publications and thus less incentive to change the system.
Allowing others to build upon your work
This aspect of the Creative-Commons licence mainly makes sense for works of art or widely used texts (such as licences), which somebody would actually want to directly build a work upon.
Building your work directly on a research article would be a very unusual thing to do, as instead of modifying the original article, you would rather publish a comment or a new article citing this article. Do not forget that building upon the ideas of an article is allowed anyway (unless they are patented), it’s just building on the text or the figures, which would be changed by putting the article under a Creative-Commons licence. Finally, it would arguably be more harmful than useful, if anybody could publish an altered version of your paper, as this would lead to confusion for readers of work which cites your paper.
With a review article it makes more sense to allow others to edit your article, which is the idea behind Scholarpedia.