Given that there are an endless series of open source and free software projects around. What makes companies and projects likely to chose to depend and use one of the existing ones rather than to write it themselves or possibly buy a closed-source solution instead? I'll try to answer a few of the things that might matter, and deal with how curl and libcurl relates to them.
Proven Track Record
The project needs to have been around for a while, so that external people can see that the development continues and that there is a continued interest in the project from developers and users. That bug reports are acknowledged and fixed, that it has been scrutinized for the most obvious security problems etc. The curl project started almost ten years ago, have done more than one hundred releases and there is now more developer activity in the project than ever before.
With companies and associations that "certify" others, you can get others' views on the quality of the projects.
The company named OpenLogic offers "certification" of open source software for companies to feel safer. I must admit I like seeing they've certified curl and libcurl. You can get their sales-pitch style description of their certification process here.
Of course I also like to see curl going to rung 2 on the scan.coverity.com list as it would mean a second (independent from the first) source would also claim that there's a reasonable level of quality in the product.
If they did it so can we
With a vast list of existing companies and products that already are using the project, newcomers can see that this and that company and project already depend on this, and that fact alone makes the project even more likely to be a solid and trustworthy choice.
Being the answer when the question comes
Being known is important. When someone asks for help and guidance about what possible solutions there are to a particular problem, you want a large portion of your target audience to know about your project and to say "oh for doing X you could try project Y". I want people to think libcurl when asked a question about doing internet-related transfers, like HTTP or FTP.
This is of course a matter of marketing and getting known to lots of people is a hard thing for an open source project with nothing but volunteers with no particular company backing.
Being a fine project
Of course the prerequisite to all points above is that the project is well maintained, the source is written in a nice manner and that there's an open and prosperous community...