It's hard to answer this question without specifics, but it strikes me as very important to differentiate between an accident or mistake in the sense of "oops, I did that by unintentionally" versus misconduct, as in "this was against policy and I deliberately did it anyways" regardless of whether you knew about the policy or had a good reason to do it or not. What you did was misconduct. You didn't accidentally email the material to yourself, you did it on purpose. Regardless of what word you use when you disclose what happened, understanding that difference, owning up to it, and showing how you've changed as a result is your best hope of gaining future employment.
Passing it off as a mistake, or trying to portray ignorance (in the sense of saying "oh, I didn't realize it was wrong when I did it") is just going to make it sound like you don't bother understanding or following policies. An employee who doesn't know about a policy important enough to fire someone over is just a ticking time bomb to an employer. That's why employers ask employees to sign technology use policies, as it appears yours has - to ensure the employee is aware of policy.
Whether or not you knew about the policy upfront, you need to be ready to discuss steps you take to stay informed about policies and ensure you're following them. You need to be ready to show that you understand that you have responsibility to understand and comply with policy, and that you're willing to do that. That brings us to your questions,
But how do I explain this to show I learnt from my mistake and get a new job ?
First, you need to be able to frame what you did for yourself. Can you explain to us what you learned? How do you approach company policy in general? How to you ensure you're aware of it, and following it?
You certainly don't need to blurt out a 5 minute monologue unprompted, but you do want to be ready to answer these questions because they will come up if you disclose what happened as you intend to. We can't tell you the best way to answer, since the best way to answer is honestly and you're the only one who can give your honest answer.
Wouldn't employers just throw my application to the bin once I declare I have been dismissed for gross misconduct?
Some certainly will, especially those who are more security-conscious. You may want to target less security-sensitive industries or environments until you've worked up enough of a resume after this event to show that you're reliable.