Graph databases provide the next evolutionary step in data storage and management. They allow for a more intuitive representation of data, which will then be easier to query for a human, and also less computationally intensive for a computer.
More traditional relational databases effectively boil down to tables with ids and values, but this rarely provides enough expressivity to adequately describe the most use cases, and when looking at multiple tables at the same time, performance degrades very quickly.
More recently, NoSQL databases (documents stores, key-value stores, and others) have tried to address these shortcomings, but often they present more serious issues surrounding data quality (e.g. denormalisation).
The problem with a tabular design that datapoints are isolated, fixed in rigid datasets where values correspond to specific headings, making it difficult to provide a description of the knowledge buried in the datastore as a whole.
Any computational analysis has to reach between these datasets it an awfully clumsy and slow manner, just like a person would have to find the right book, flick between pages, search through the columns, every single time.
This is very slow, but more importantly, limits the complexity of the analysis that you can do.
How to query a graph database?
When using a graph you can consider everything at once, not just a tiny slice at a time, giving you a wholistic view of all your data with the proper context that provides. Beyond that, the web-like structure of the graph itself opens the door to more computationally complex analysis, and enables this at extreme speeds. For example, machine learning and AI are much more receptive to this kind of data, allowing you to do so much more than was previously possible.
Even without these imposing tools, the structure alone provides benefits with its human-centered design. Ultimately, even machine interactions are written by humans, and this makes fundamental analysis and complex querying far simpler for the user, allowing vastly superior solutions to be designed for otherwise impossible tasks.