What do mathematicians mean when they say some conjecture can’t be proven using the current technology? When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What do they mean by that? Are they talking about the axioms? Or are they generally speaking in terms of intelligence and mathematical abilities?

These ones are just at the top of my mind:

https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/

https://m.youtube.com/watch?v=MXJ-zpJeY3E (skip to the end where the talk about the Riemann Hypothesis)

But I’ve seen many, many more examples
This doesn't have any formal meaning.  It just means that they believe the problem can't be solved with the techniques that mathematicians have already developed and instead some big new idea will be necessary to solve it.  That is, "the current technology" refers to the collection of proof methods that mathematicians have discovered already.  It's meant as a sort of metaphor between mathematics and engineering: some engineering problems can be solved by just finding a clever way to put together already existing technologies, while others require major new inventions.  In the same way, some mathematics problems can be solved by just finding clever new uses of ideas that are already known while other mathematics problems require something more novel.
It means that the methods they have don't suffice to solve the problem and the technology they researched wont give back a mathematic answer that satisfy their conjecture. 

The problem in general (which is the amount of times “[the conjecture] cannot be solved using the current technology” is cited) its not a factual (some would say 'engineering') problem because the question is generally not a finite and descriptive one (refer to 'How do I ask a good question?' in stack overflow https://stackoverflow.com/help/how-to-ask 
 ). 

Many mathematic problems of the sort don't seek a concrete answer but rather a general solution. Meanwhile, technology works great to get concrete solutions to specific problems and therefore technology would have to be directed/tailored to solve that specific problem or to give broader answers that can answer such question, which is something not done yet for the problem in question.