We're all familiar with Matthew 13 where Jesus delivers the parable of the sower. He gives us the four possible conditions/reactions that we will encounter when the Gospel is preached. Not only is the teaching itself profound, but in verse 10 it leads the disciples to then ask Jesus his purpose for parables, to which Jesus responds, to you it has been given to know the secrets of the kingdom of heaven, but to them it has not been given (ESV). I personally cannot see how else this verse can be interpreted? I think its overtly clear that Jesus is telling his disciples that they have received the ability to understand what was being spoken, and the others have not; which, as Jesus goes on to say, is a fulfillment of Isaiah 6. So, I think it's safe to conclude from this passage (and others) that the ability to [ultimately] hear what Jesus [truly] meant was something that was/is not given to everyone.
Now why would Jesus say that the revelation of the Gospel has been given to his disciples and not given to the others, outside of teaching the doctrine of election? We see the same thing taught in several other places, particularly John 10:26. For purposed of keeping this post short, the classical Arminian appropriation will not be considered. I will say this: Why would the biblical authors use concepts like this, and terminology such as "chosen," "elected," "given/not given," if in fact something wholly other was intended? For purposes of clarification, and to avoid confusion it would seem much easier to leave the aforementioned out.
I'm going to end here and give you John Calvin's commentary on this verse. I think captures the essence of the passage by stating,
To ascertain fully the meaning of the present passage, we must examine more closely the design of Christ, the reason why, and the purpose for which these words were spoken. First, the comparison is undoubtedly intended by Christ to exhibit the magnitude of the grace bestowed on his disciples, in having specially received what was not given indiscriminately to all. If it is asked, why this privilege was peculiar to the apostles, the reason certainly will not be found in themselves, and Christ, by declaring that it was given to them, excludes all merit. Christ declares that there are certain and elect men, on whom God specially bestows this honor of revealing to them his secrets, and that others are deprived of this grace. No other reason will be found for this distinction, except that God calls to himself those whom he has gratuitously elected. (my italicizing)