The security dilemma is like a game you might play with your friends. Imagine one of your friends has a toy and they don't want anyone else to have it. In order to protect it, your friend gets a big, strong dog and ties it to the toy. That way, if someone tries to take the toy, the dog will bark and hopefully scare them away.
But someone else might see your friend's dog and decide to get their own dog to protect their stuff. Then you and your friend might both be worried that the other person's dog might attack - even though neither of you is trying to hurt the other one. This is the security dilemma - two people trying to protect themselves from each other, but in the process making the other person more worried. In the international world, this might look like two countries trying to get bigger militaries to protect themselves from each other, even though neither one wants to attack the other one.