What it means:

unlikely alliance

How Shakespeare used it:

When Trinculo seeks shelter from a storm under the cloak of a creature he's very unsure about – he wonders if it's a man or a fish – he comments "misery acquaints a man with strange bed-fellows." (The Tempest, Act 2, Scene 2)

Modern example:

"But there's another key reason Philip Morris lobbied hard for FDA regulation, aligning itself with strange bedfellows like the Campaign for Smoke-Free Kids [and] the American Lung Association ..." — Kate Pickert, Time, June 12, 2009