An inquiring mind
Great exploratory testers want to find out everything. They want to know what's behind this door (or button) or and what's hiding in this box (or feature). It's very important for an exploratory tester to be inquisitive and interested, because they're on their own when it comes to checking the different parts of the website or application. Exploratory testing doesn't pre-determine what's to be tested or how, so it's up to the testers themselves to investigate the nooks and crannies.
A competent extemporaneous test designer
Exploratory testing requires testers to figure out what kinds of tests they want to run themselves based on what they discover. Testers need to be competent test designers who can roll with the punches: if an investigation of a specific part of the software turns up an error, testers need to be able to improvise new tests that look for related problems and issues. By being good at coming up with impromptu additional tests, exploratory testers can determine the scope of a bug, or at least circumscribe the impact of it, which makes the task of fixing it for a developer easier.
An observant viewer
Being observant is also correlated with the first characteristic of an inquisitive mind. Exceptional exploratory testers need to be not only curious about the product, they also need to be able to notice the aberrations and document them. This also includes "observing" settings, environmental factors, and internal mindsets as well. All of these need to be written down for an excellent exploratory test. It's a holistic approach to testing, while also being more liberating for the tester themselves. The tester needs to consider and record anything that may have had an influence on the test. When testing in the real-world, this could be something like wifi vs cellular, signal strength, battery level, or even ambient noise. What are the settings on the test device? Some of this is easily recorded, some requires additional investigation. And because we're dealing with human beings, it can also be important to note what the intentions and mindset of the tester was.
A communicator par excellence
Product and project managers extend quite a bit of trust when they decide to use exploratory testers. That means exploratory testers need to be great about communicating what they've found. Because there's no separate test designer sitting down with the product manager to pre-determine individual testing steps and expected outcomes, it's necessary for testers to be able to communicate the idea behind a given test, the circumstances surrounding it, and then the outcomes that result from it.
The best exploratory testers know when to use words, when to use screenshots, and when to take videos. They take into account the point of view of the people who will be using their work: developers, QA professionals, and product managers and offer the key information those people need to do their work.