Imagine that your neighborhood is organizing a Treasure Hunt where some house has a key nailed to the inside of the door. To win, all you have to do is find the house, take a photo of the key (it's nailed ;) and return back with your photo. It doesn't matter when you get back (you don't have to be first), so as long as you get back, you win.

With regards to SSH, a private key tells you the address of the house that has the key, so you just have to hop on your scooter and ride over there. Your competitors may have a bunch of people and Bugattis, and as long as they find the house with the key, they'll be able to decrypt your SSH session or PGP message.

Since we want to make it really hard for our competitors to find the key-house, we should expand the game to include every house in the world. Granted, we have to go scootering for a bit to decrypt things even with the private key, but it'll take much longer for the competitors to find the key-house. And that's exactly what increasing the bitlength of your key does.