You melt the glass in controlled manner and let it to fuse together. As far as the process goes it is mostly similar to how lab technicians in chemistry labs build custom glassware, but with much smaller dimensions and better control

In practical terms you use ridiculously expensive computer controlled portable machine that just does it without much of manual intervention. It is done by such automatic splicers almost since fiber optics are used for practical applications. There were manual (mostly mechanical) splicers that are not too hard to use (it certainly requires less skill than above mentioned laboratory glassware), but it's more economical to just automate the task.

The problem is that even almost perfect splice incurs significant additional signal loss, so for long cable runs you do want to do smallest number of splices (and connectors, which are orders of magnitude worse) as possible.